He tries counting sheep to fall asleep, as per cliche, but his mind won’t shut off and he starts free-associating. Square root of nine for three, then IV for four…which leads to a centurion sheep leading a full unit.
That looks like more than just one unit, if the “148” is an accurate count of the “flock”.
My favorite rock concert album and video were “Velvet Underground Live MCMXCIII”. Which devotees know to say aloud as “em cee em ex cee aye-aye-aye”.
You know what year that was done, too.
In the first panel, it’s clear he’s counting sheep to get to sleep.
In the second panel, you see the square root of 9 (which is 3). Here’s where the reader sees that more creative ways of counting are being introduced.
The third panel has two parts: The first part shows “IV” (the Roman numeral for 4). Along with the sheep in Roman regalia, this goes along with the trope introduced in the second panel, in that more creative ways of counting are being used.
But then there’s the third panel’s second part: “CXLVIII” and it being represented with a legion of sheep in Roman soldier garb. That’s kind of confusing, so what’s that all about?
I think it’s just a visual pun that showcases the difficulty of Roman numerals. Let’s face it, once you get to “IV” (I plus V is… 4?) things get a little wonky in the Roman world of numbers. (What’s after “IV”? IV + I, right? So, “IVI”?) And if “IV” isn’t confusing enough for you, I’m sure that “CXLVIII” certainly is.
Personally, I think that the strip would be slightly clearer if the “IV” sheep was placed in its own separate panel (the third panel), and the “CXLVIII” sheep were placed in the final fourth panel. That way, the creativeness of counting sheep is established in the first three panels, and the confusion of Roman numerals is kept separate in the fourth panel.
In case any reader here is relatively unfamiliar with the Dark Side of the Horse comic, the “counting sheep to fall asleep” trope is one of those he reuses frequently and takes joy in twisting into unexpected variants. Maybe second only to “elevator triangular call buttons” and slightly ahead of “highway roadside hazard warning signs”.
Most kids learn Roman numerals in school, but my knowledge of them was perfected by watching “Looney Tunes” every Saturday morning. Even as kids, we observed that the copyright date in the opening credits was a fairly reliable indicator for the quality of the upcoming cartoon. The best ones were dated 1951-53 (MCMLI to LIII), anything before that was usually too corny, and anything 1956 or later was usually garbage (with some notable exceptions).
P.S. Chuck Jones was the most reliable director, but an even more reliable indicator were stories written by Michael Maltese.
P.S. @ J-L – There are some exceptions, but DSotH is almost always a three-panel strip.
” Let’s face it, once you get to “IV” (I plus V is… 4?) things get a little wonky in the Roman world of numbers. (What’s after “IV”? IV + I, right? So, “IVI”?) And if “IV” isn’t confusing enough for you, I’m sure that “CXLVIII” certainly is.”
Um…. no…. I always considered Roman Numerals to be very straight forward, a number to the left of a larger means to subtract from so IV must be read as V- I which is ….. 4? well, yeah….. and IVI = V-I+I=V. Although there are semantic rules of cleaning and not going exceeding magnitudes in subtraction… such thinking is useful if you try to add/subtract/multiply roman numerals without decimal conversion[*].
And CXLVIII = 100+(50-10) + V+III = 148. What on earth is confusing about that?
[*] FOUR + ONE = FIVE because FOUR = (FIVE – ONE) and so (FIVE -ONE) + ONE = FIVE. The -ONE and +ONE cancel. IV +I = IVI, THe I’s before and after cancel. Other example: LXIV +XCVII = XCLXIVVII. The Xs one each side of CL cancel and single I on each side of VV cacel so = CLVVI. The VV combine to X and =CLXI. In other words 64 + 97 = (50+10 + (5-1)) + (100-10 + 5 + 2) = 100+50 + 5+5 + 1 = 100 + 50 + 10 +1 =161. Simple.
I dunno though. Dark Side of the Horse set it’s own level for sight puns that simply having 3 = square root of 9 and that’s a nerdy thing so we have a sheep in glasses, and a roman sheep leading a legion of arbitrarily 148 sheep just doesn’t seem to cut it. Unless there is some significance to 148.
Although there are semantic rules of cleaning and not going exceeding magnitudes in subtraction…
Woozy, yes, these are interesting to contemplate. I like some awkward examples that eschew even single subtractions, and will show four of a kind for some symbols. Going the other direction, it is disconcerting to see someone (mostly moderns) try to subtract across multi levels. The recent practical occasion for needing to try this out was of course the year 1999. Would you accept “IMM”? Or maybe “MIM”? Many stuck with “MCMXCIX”.
heres what I should have said.
VI is ONE after FIVE. So that is obviously SIX.
And IV is ONE before FIVE. So that is obviously FOUR.
You will occasionally see in (in tarot cards mostly) VIIII for nine, and occasionally you will see on clocks IIX for eight.
Once the nineties occurred and we sometimes saw MCMXCIII I always figured the Cs cancel and it should be MXMIII but I guess some people think substraction should only be one degree of magnitude. I get it but ….
Actually in the early 2000s I saw a date MCMCI and I figured someone after a century of MCM + (I to XCIX) just couldn’t break the habit.
Mitch, I think those of us using digits would thing 19 = 20 -1 would first think 1999 could be IMM but psuedo romans would think 19=(10 + 10) – 1 = 10 + (10-1) and do MIM. Of course real romans think in magitudes and an 1999 = 1000 + 900 + 90 + 9 so MCMXCIX but from a calculating point of view those cross cancelinng C’s and X’s are …. wow.
I agree that he’s obviously counting sheep to fall asleep, that he’s doing it in an odd way, and that Roman numbers can be confusing. But I assumed there was some significance to CXLVIII or 148. If the point was that Roman numbers can be confusing, it would make sense if IV was followed by IVI or something. But even someone confused about Roman numbers wouldn’t try to follow IV with CXLVIII.
Fun fact (which I’m sure most of you already know): on clock faces that use Roman numerals, at least on the old ones, 4 is represented as IIII.
I sometimes use Roman numerals to impress my students with just how easy math can be, and how much harder it can be. I usually also point out that the numbers they’re used to are Arabic numerals, because Arabians came up with lots of clever stuff.
@ Chak – The reason they do that is not for mathematical simplicity (nine is always “IX”), but for visual symmetry: the width of the “IIII” on the right harmonizes better with the “VIII” (eight) on the left.
BTW, I got the Bliss/Martin book! Hefty physical book. Now to go relax and read …
“the numbers they’re used to are Arabic numerals, because Arabians came up with lots of clever stuff.”
They did come up with lots of clever stuff. But much of what we label “Arabic” is only because it came to Europe via Arabia. Positional representation of numbers was invented by Hindus in China and/or India.
Reading Woozy’s explanation of Roman numerals brought to mind this quote from Johann Sebastian Bach: “There’s nothing remarkable about it. All one has to do is hit the right keys at the right time and the instrument plays itself.”
If you have a chance, visit the “History of Philosophy (without any gaps)” website / podcasts / books. A now-concluded sequence within the still-ongoing main Western thread was “Philosophy in the Islamic World” which was quite an eye-opener for me. https://historyofphilosophy.net/series/islamic-world
Becky/woozy: I started to write something about how Roman numeral rules are not straightforward or obvious, but then gave up, because how do you convince someone that rules they think are obvious are not obvious?
Webb (Gordon Ramsey): “[Cooking this dish is easy.] It’s just local ingredients, simply cooked.”
Mitchell: “By you! King Lear is just English words put in order.”
I think I’ve mentioned this before, but I used to be a math major, and we did actually have conversations like this: “This step is trivial/obvious.” “I’m not sure, it is really obvious?” [Half an hour of consulting definitions, sketching proofs, etc. . .] “Oh yes, it is obvious, I see it now.”
If he jumps (!) from 4 to 148, it’s because the next 144 (=12*12) sheep come in a flock/legion instead of one by one, as is customary.
Now I can’t remember where, but I was going to bring up exactly this joke a couple days ago! Well, my version has the instructor / lecturer at the board say “This step is obvious … wait a minute ..” and check for several boards worth of space and most of the remaining time, to finally conclude “Yes, it is obvious”.
Oh now I remember! It was when everybody was yelling at me for calling the Frog Applause pun “straightforward”. I was ready to try to prove it! :-)
Interesting discussion, but I think dvandom had it at the first comment. I can relate to going to sleep peacefully and then the mind takes over.
“All one has to do is hit the right keys at the right time and the instrument plays itself.”
Rosenkrantz and Guildenstern would beg to differ:
Ham. I pray will you play vpon this pipe? Ross. Alas my lord I cannot. Ham. Pray will you. Gil. I haue no skill my Lord. Ham. Why looke, it is a thing of nothing,
T’is but stopping of these holes,
And with a little breath from your lips,
It will giue most delicate musick. Gil. But this cannot wee do my Lord. Ham. Pray now, pray hartily, I beseech you. Ros. My lord wee cannot. (me? Ham. Why how vnworthy a thing would you make of
You would seeme to know my stops, you would play vpon [G1]
You would search the very inward part of my hart, mee,
And diue into the secreet of my soule.
Zownds do you thinke I am easier to be pla’yd
On, then a pipe? call mee what Instrument
You will, though you can frett mee, yet you can not
Play vpon mee, besides, to be demanded by a spunge.
And I thought: Why does Shrug’s copy of Tom Stoppard have the dialogue in such old-fashioned spellings?
But roman numerals are easy.
To read them: I = 1; V=5, X=10; L=50;C=100;D=500; M=1000 and putting an overline increases a symbol by a magnitude of a thousand. They are list higher left to right by highest denominations first and you add them up. Because they are listed highest to lowest if you ever do see a lower denomination before a higher denomination that means the lower denomination is subtracted.
So CXLVIII = 100 + (50-10) + 5 + 3=148
That’s it!
To write them there some intuitive but hard, but not too hard, to formulize rules. If you can use larger denominations do so (Represent 150 and $CL and not LLL or CXXXXVV because the largest block you can use is C so you must use it). The primary units are I, X,C, M of magnitudes 1, 10,100,1000. V,L,D are “half units” magnitude 5x 1, 5x 10, 5x 100. Their purpose of existence is to allow us to represent five to nine units of a magnitude in fewer terms. To allow us even fewer terms placing a primary unit if the magnitude immediately less then the current primary or half unit, to the left of a unit (so we can think of it as subtraction) allows for us to express $4$ or $9$ of the lesser unit.
If he jumps (!) from 4 to 148, it’s because the next 144 (=12*12) sheep come in a flock/legion instead of one by one, as is customary.
And for me that’s the missing step!
If a legion is 12 x 12 rather than the 10x 10 as I assumed and if Horace is keeping a running tally (which hadn’t occurred to me but is clear in hind sight that he would be) then that explains the 148.
And that is enough to make a joke (whereas an arbitrary roman numeral would not).
But that’s gross!
Yes, I agree that if Roman legions came in groups of 12*12 that would make the joke make sense. But Olivier, is that actually the case, or were you just speculating that they might come in that size? I didn’t find any indication that that was a standard Roman legion size with a quick search.
According to a handy dictionary, a Roman legion was “A division of from 3000 to 6000 men (including cavalry) in the Roman army”.
Interesting article relating to Roman numerals and mathematics, and how the Roman numeral notations had changed.
I don’t think the “148” that Samson picked has any “meaning”: it was probably just selected to make the number sufficiently complex, without being unreasonably large. As noted above, it is far too small for a “legion”, but also too large for the “century” that a “centurion” would command. The size of these “platoons” varied considerably (from 30 to 80, depending on the era), but never reached 100, let alone 144 soldiers.
P.S. @ WW – Back in college, one alleged strategy for solving math problems was to work down from the top equation, then work up from what the solution was known to be, and then write in the middle (between the two parts): “…it is thus intuitively obvious that…”
P.P.S. One guy who actually tried this trick got a note back from the T.A. that said “It is to me, but I don’t think it was to you.” (with points deducted).
Still trying to figure out if 148 has any significance here, or if it’s just a number that’s comically complicated to express in Roman numerals. I like the “144 more” idea but I don’t know if that’s a specific Roman unit size either.
Re: clocks with IIII for 4: I was always told that it was a superstitious avoidance of the letters “IV” which could refer to IVPPITER. Seems to be an urban legend, since the “subtractive” notation appears to be a medieval innovation that didn’t come along until centuries after anyone cared about Jupiter except perhaps as a planet.
A baker’s dozen is thirteen; a regular dozen with one more.
A gross is 144; a dozen dozens.
So what is a baker’s gross?
Answer: chicken guts in the doughnuts.
Kilby: I suspect that explicitly writing “it’s intuitively obvious,” or “it trivially follows” is actually more likely to call the T.A.’s attention to that line, and make them question if you know what you’re doing, than just saying “therefore” or “it follows.”
Yes, and I think we’re onto something with these thoughts about “a gross”.
All right, he’s asleep or hypnagogic, and logic is reduced. So the appearance of Roman soldiers ties into the Roman numerals, but doesn’t oblige the narrative to make them a legion or a century or anything Roman. Instead, Horace reverts to his sometimes business persona, and puts in an order for a common commercial measure of additional sheep, one gross. They are arriving in one bunch, and his running total reaches the number shown.
There we are, Cello! Sorry, I meant Viola! Sorry, I meant Wah-La!
On “Obvious” — At one time I worked for the Post Office. We trainees were given overviews of some matters we might or might not be dealing with on a daily basis, including various regulations. One set of rules was about how the station and individual carriers and mail handlers needed to treat various kinds of mail. Some important rules were about First Class mail of course. A different set protected Third, or Parcels, “of obvious value”.
Mailrooms and shippers knew all about postal regs. To make sure their Third Class and Parcel mail was handled safely and promptly, they wanted to ensure these pieces would count as being “of obvious value”. Some of them had a big rubber stamp (standard issue from stationery stores) and marked their mail .. wait for it … no not “VALUABLE” which you’d think would make it obvious … but instead “OBVIOUS VALUE”.
So this discussion has made me realize that you can mathematically prove that an MD is worth more than a JD:
MD = 1500
JD = ID = 499
QED
Since we’re 41 comments in with no clear explanation of why 148 specifically was chosen, I’m guessing that it’s just a randomly chosen number. If it was another comic, I would have accepted that much sooner, but with Dark Side of the Horse I have more of an expectation of an “Aha!” insight after a potentially very long wait.
So you don’t accept the “got to IV by single counting, then ordered another gross from the sheep supply — 4 + 144 = 148 = CXLVIII”?
Oh, sorry, I actually thought you were joking! :o
That seems pretty convoluted to me. There’s no sign that he’s resorting to a business persona, and why would the 144 sheep that he ordered be carrying a Roman banner (and wearing uniforms, I think)? I think it’s more likely that this is just a bunch of Roman sheep following their leader, and that 144 is just a randomly chosen large number.
” If it was another comic, I would have accepted that much sooner, but with Dark Side of the Horse I have more of an expectation of an “Aha!” insight after a potentially very long wait.”
Exactly!
“Oh, sorry, I actually thought you were joking!”
Me too but I thought he was joking about stuffing donuts down his pants too.
larK says: So this discussion has made me realize that you can mathematically prove that an MD is worth more than a JD: MD = 1500 JD = ID = 499
But with that prompt we cannot forget to remind everyone of the famous point that Halloween and Christmas are just notational variants:
31 OCT == 25 DEC
The problem with Roman legions is that their composition has varied throughout history.
Roughly, one legion is made of 10 cohorts, each cohort holds 3 or 4 maniples comprising 120 to 160 men. One maniple is 2 centuries.
In the Straight Dope article, they mentioned sundials. Indeed, examples of ancient Roman sundials did use IIII so it’s possible that early clock makers copied that style.
Following from Olivier’s hint:
For IV we have one centurion. That might be followed by two centurions. Instead, we have 148. A common size of a maniple (two centuries of 80) is 160. In the earlier panels, we have 1 + 2 + 9 = 12, which combined with 148 makes a those two centuries (instead of two centurions).
Q.E.D. Simple and straightforward :)
A more interesting question to ask would be whether there is some significance to 148 in Finland, the cartoonist’s home country.
Q.E.D. Simple and straightforward
Intuitively obvious to the most casual observer. And they don’t get much more casual than this observer, amiright people?
It’s a nice theory, except that in the earlier panels, we have 1+2+√9=6, and not 12.
Ah, but the panels are square, and there are three of them, so you get 6^2/3… 12!
I’ve been trying to find a modern military unit of 144 men ; the range would make it a company, but there are huge variations here as well.
Actually, if you want 144, you can buy plastic toy soldiers: it seems they’re sold by the gross.
@ Mitch4 – “Halloween and Christmas are just notational variants: 31 OCT == 25 DEC”
Does that mean that the website subheader for next month will read
“It’s OctEmber!” ?
P.S. After all (as Churchill LaFemme once observed), Christmas falls on October 86th.
And as I think I’ve mused here before, “740” used to be The Number of the Beast, before the Beast’s agent took our his ten per cent commission.
Could this be a reference to Caesar’s 5th legion?
Mitch4 –
So, is the fact that my mail is not stamped “obvious value” why instead of forwarding (from our PO Box) some of it they are returning some of it to the sender (NYS Tax and a credit card company) or it just disappears (16 bank statements for us, our business, and the two clubs of which I am treasurer)in 3 months) – and then when we got desperate enough to make a late night trip to the PO mid corona virus stay at home – there was mail (first class again) sitting in our PO box?
Worst of all we were having our mail go to our PO box – at a post office other than the one which delivers to our house – because we had been having a problem with mail going astray and not getting to our house, so other than the returned mail and what was in the box when we went, we have no idea of which post office (our home post office or our PO box post office) is the one losing our mail. I am currently waiting for 2 checks to come from various places which are long over do since mailing plus another check had to be replaced as it never made it here.
I so much miss and appreciate the daily trips to our PO Box for mail in and mail going out.
Recently read an article in BBC History magazine that explained Roman Numerals were taken from the Etruscan system, They used different symbols which were more complicated, but they receive no credit for their system being used to develop the Roman system.
Ay ay ay, who knows what is up with the mail these days! But my experience with the “obvious value” stamp was from the summer of Apollo 11 and Woodstock, so things may have changed by now, as they say on the NPR politics round-up podcast.
@MArk H.: how so?
“but they receive no credit for their system being used to develop the Roman system.”
Well, considering the general impression is that the Roman numbers are unyieldy and a poor choice for calculations (I’ve only been arguing Roman numerals are straightfoward and not that bad; I’m not arguing that they have any advantage) I don’t think many people think that there is much credit deserved.
(After all, if they hadn’t used Etruscan numbers they’d have used something else…..)
(Actually, just how many possible number systems are possible, really?)
@ woozy – How many? I’m not sure whether I’m up for a discussion about all the multitudinous varieties (meaning magnitudes) of “infinity”. I’ve got one kid who has gotten as far as “fractions” in school (just adding and subtracting, they recently started multiplying them, but division is a long way off). He’s aware of negative numbers, but he picked that up from me, they haven’t touched them in school yet. I’ve already warned him that he will have to get used to several more massive expansions to the “set of known numbers” (all he has right now are natural numbers and a dusting of rationals, there’s still integer, real, irrational, and complex numbers, and probably several other types that Winter Wallaby could add).
And just wait until you get mugged by the ordinal infinites. So much more structure than the cardinals have, even with their surprises.
One of my students was taking a course to prepare her to teach math to k-8 children (dog help those kids, but that’s another story) and she was learning (sorta) about various historical number systems. The one I particularly liked was base 60. It was fun, for me at least.
It was fun for a minute. Or a second-minute.
Well, just to a degree.
@ Chak – I took a “History of Math” class in college that did a similar sort of review. At least for the Babylonian stuff all we had to do was learn the mathematical notation, the professor didn’t expect us to learn any matching vocabulary (like he did for Greek).
P.S. @ Danny Boy – LºL ;-)
Speaking of modular or Base arithmetic, one of the oddest but still workable ones I’ve seen is what I recall as “Base-3 subtractive” but this Wikipedia article titles “Balanced Ternary”. Like “normal” ternary, it has three digits, but instead of {0,1,2} they are {-1,0,1}. (In the article, and generally, a letter is used for -1, here T.) As their chart shows, you can represent and count to numbers in a predictable and regular way, though maybe hard to do in your head.
The article also shows how to do … not “decimals” I guess but fractionals written with digits to the right of a radix point.
I was surprised at how old the article says the system is. I thought I must have learned of it from something like a Martin Gardner column, and that it was credited to somebody like Conway, and was especially suited for representing the solutions or strategy of some puzzle or game.
Horace never gives up on sheep-counting. Even when reduced to more-usual numerals.
@ Mitch4 – I thought that the name “Conway” sounded familiar, but only after I looked him up did I remember that he invented the “game of life”. I’m sure there are tri-state variations of the game in which the cells might be described as “live”, “dead”, and “sick” (or some other amusing adjective).
Conway was on my mind after recently hearing the special memorial edition of BBC More or Less, linked here . The posted summary: “Mathematician John Horton Conway died in April this year from complications related to Covid-19. We remember the man and his work.” Some of it is about him ruefully accepting always having Cellular Automata (Game of Life) mentioned first among his achievements.
John Horton Conway FRS (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life.
I think Stephen Wolfram would object to cellular automata being dismissed as “recreational mathematics”…
Ditto John Conway.
I wonder if they would really consider that dismissive. I’m not a mathematician (although as a physicist I did research in some cellular automata models) and for me the “recreational” adjective means that it’s particularly awesome, rather than excluding the possibility of usefulness.
Right, in the Wikipedia article take note of the section on Conway’s friendship with Martin Gardner, the longtime “Mathematical Games” columnist at Scientific American magazine and a key figure in the popularity of recreational math as a recognized hobby or amusement , attractive to many nerdy ordinary people but also to serious mathematicians.
Mitch4 wrote: “Some of it is about him ruefully accepting always having Cellular Automata (Game of Life) mentioned first among his achievements,” and the Wikipedia article kind of dismisses it as an additional bit of trivia; meanwhile, Steven Wolfram puts cellular automata front and center in his New Kind of Science. Now, it’s not clear whether Wolfram is really on to something or not, but if he is, then cellular automata just might be the most important thing Conway ever did…
I appreciate that real math geeks don’t see “recreational” as a disparagement, but then they have interesting takes on common words (eg: trivial) that most people wouldn’t properly appreciate…
The true math geek in me finds the word “recreational” redundant. What field of math isn’t recreational?
I worked my way thru much of Winning Ways and enjoyed it very much, even when getting beyond my depth with the surreal numbers. It’s also a lovely physical book, with even some color printing so diagrams can easily distinguish stuff by putting things in red and blue. If you can get hold of it, I bet you would enjoy it too.
The mentions of Conway working on “the theory of games” is not the same thing as “game theory” in the sense of von Neumann & Morgenstern and then “Nash equilibrium” and all that stuff. No, it’s games like Nim. (These are the subject of Winning Ways and On Numbers and Games .)
Do you know Nim? Shall we play a couple rounds? Warning: it is a solved game. There are winning positions and losing positions.
You put down on the table several rows of some sort of token. Players take turns, on each turn removing one or more tokens from a single row of the layout. You cannot “pass” and take away none. You cannot take away from more than one row. The player who is able to take the last remaining token and clear the table is the winner. (It is often played the opposite way — the player “forced” to take the last token loses. The analysis is the same, in that “safe” and “losing” positions along the way are actually the same, up until you have to switch at the end.)
Let’s use the classic “Marienbad” layout, of 1-3-5-7.
1 * 3 * * * 5 * * * * * 7 * * * * * * *
This is a winning position (to leave). So to be fair, I will go first, taking some away, and leaving a losing position (to leave) — which your move can convert to a winning position.
I will take 2 from the row of 7:
1 * 3 * * * 5 * * * * * 5 * * * * *
Now it’s up to you …
To clarify the relationship of winning and losing positions:
— If one player leaves a winning position, every possible move by the opponent leaves a losing position
— If one player leaves a losing position, there exists at least one move by the opponent which leaves a winning position
The “solution” for winning play is a formula or algorithm for finding the move, presented with a losing position, which will produce a winning position. Once you have left a winning position, if you use the formula or algorithm, you can continue to receive losing positions from the opponent and convert them to winning positions.
Mitch4: I play the Calvinball gambit, of changing the rules absurdly to something like “But then the universe declares the tallest player wins.” Or “suddenly Hulk Hogan appears in a puff of smoke and says that the player with the longest muttonchops wins, Brotha.” Looks good for me.
Your turn.
Mitch4: I loved that book. The analysis of how to play Dots-And-Boxes was amazing.
Shrug: A strange game. The only winning move is not to play.
For a nice take on the use of ordinary words for special ideas in math, look up ‘Finite simple group of order two’ by the Klein Four group on YouTube. I’d give a URL but I can’t right now.
@ Chak – I found the link to the “Finite Simple Group of Order Two”, but I think most of the humor will be lost upon non-math majors.
@ WW – I was quite amused by Shrug’s permutation, precisely because it led to your “winning” move. Even though Mitch4 gave a very precise description of the game, I cannot do the binary arithmetic for the four row layout in my head (I know all the winning patterns for the 3-4-5 layout, so that one doesn’t require any “work” for me). In addition, I vastly prefer the “last move loses” version.
You don’t have to do it in your head, you can do it on your fingers! Right pinky is ones, right ring is twos, right middle is fours, right index is eights …
To evaluate the original 1 3 5 7, the 1 gives you pinky down.
The 3 reverses the twos and the ones, so it’s ring down and pinky up.
The 5 is the middle and pinky down, leaving last three fingers down.
The 7 is last three fingers, so they come back up.
Leaving no fingers down, so it is a safe position, and any move leaves unsafe.
For my leave of 1 3 5 5 you could also start with the fingers, But easier to note that the two 5 5 cancel , so we have 3 and 1, which reduce to a 2. The only 2 in that board is in the 3 (not in the 5s!) so we will take 2 from the 3, leaving
1 *
1 *
5 * * * * *
5 * * * * *
which, almost by inspection is a winning position to leave. We will just match whatever the opponent does, in matching row.
It is not yet at the point where strategy changes for Kilby’s preferred misêre version, as they are called.
“I vastly prefer the “last move loses” version”
But that makes absolutely no difference in evaluating this position, or almost any position well short of the endgame.
I don’t carry the 3 4 5 positions in my memory, but it is quick enough on fingers or simply mentally to calculate the Nim Value (as they call it) to be 2. So take 2 from the 3, leaving 1 4 5. And again, just as we could see the pair of 5 5 cancelling each other in the other example, this is only slightly harder to see as 1 plus even n plus odd n+1, a common 0 grouping.
And even though small, this is still not quite endgame, and makes no difference between normal play and misère version.
@ Mitch4 – Cool, I like the “binary finger arithmetic” algorithm. I’ll try to teach that trick to my son.
Thanks for the link, Kilby.
Cool, I like the “binary finger arithmetic”
Back when my joints and muscles had more agility, and I more often found myself in waiting-around situations without amusements, I would transform the “tapping your fingers” nervous gesture into binary counting. I maintained I could reach 1024 error-free but I was probably kidding myself.
I used to know all sorts of math – but being an accountant – the basics of arithmetic, estimating, and trying to figure out what the client’s numbers mean is about all I deal with these days. I used to be much better at doing arithmetic in my head, but since I started using a calculator and then a computer much of it has left my head so I have room for less important information – such as why my colonial self and her mother will never actually “own” the family property, just get to use it for life with my nephew ending up owning it after we are gone, etc. These days I am glad if I can figure out what day it is without looking at my day of the week pill box.
I suppose he qualifies, under the meaning of “typical male citizen relaxing at home”. But he doesn’t seem to actually have vi of anything going on, just that i cup, and i jug.
Comics I Don't Understand 
He tries counting sheep to fall asleep, as per cliche, but his mind won’t shut off and he starts free-associating. Square root of nine for three, then IV for four…which leads to a centurion sheep leading a full unit.
That looks like more than just one unit, if the “148” is an accurate count of the “flock”.
My favorite rock concert album and video were “Velvet Underground Live MCMXCIII”. Which devotees know to say aloud as “em cee em ex cee aye-aye-aye”.
You know what year that was done, too.
In the first panel, it’s clear he’s counting sheep to get to sleep.
In the second panel, you see the square root of 9 (which is 3). Here’s where the reader sees that more creative ways of counting are being introduced.
The third panel has two parts: The first part shows “IV” (the Roman numeral for 4). Along with the sheep in Roman regalia, this goes along with the trope introduced in the second panel, in that more creative ways of counting are being used.
But then there’s the third panel’s second part: “CXLVIII” and it being represented with a legion of sheep in Roman soldier garb. That’s kind of confusing, so what’s that all about?
I think it’s just a visual pun that showcases the difficulty of Roman numerals. Let’s face it, once you get to “IV” (I plus V is… 4?) things get a little wonky in the Roman world of numbers. (What’s after “IV”? IV + I, right? So, “IVI”?) And if “IV” isn’t confusing enough for you, I’m sure that “CXLVIII” certainly is.
Personally, I think that the strip would be slightly clearer if the “IV” sheep was placed in its own separate panel (the third panel), and the “CXLVIII” sheep were placed in the final fourth panel. That way, the creativeness of counting sheep is established in the first three panels, and the confusion of Roman numerals is kept separate in the fourth panel.
In case any reader here is relatively unfamiliar with the Dark Side of the Horse comic, the “counting sheep to fall asleep” trope is one of those he reuses frequently and takes joy in twisting into unexpected variants. Maybe second only to “elevator triangular call buttons” and slightly ahead of “highway roadside hazard warning signs”.
Most kids learn Roman numerals in school, but my knowledge of them was perfected by watching “Looney Tunes” every Saturday morning. Even as kids, we observed that the copyright date in the opening credits was a fairly reliable indicator for the quality of the upcoming cartoon. The best ones were dated 1951-53 (MCMLI to LIII), anything before that was usually too corny, and anything 1956 or later was usually garbage (with some notable exceptions).
P.S. Chuck Jones was the most reliable director, but an even more reliable indicator were stories written by Michael Maltese.
P.S. @ J-L – There are some exceptions, but DSotH is almost always a three-panel strip.
” Let’s face it, once you get to “IV” (I plus V is… 4?) things get a little wonky in the Roman world of numbers. (What’s after “IV”? IV + I, right? So, “IVI”?) And if “IV” isn’t confusing enough for you, I’m sure that “CXLVIII” certainly is.”
Um…. no…. I always considered Roman Numerals to be very straight forward, a number to the left of a larger means to subtract from so IV must be read as V- I which is ….. 4? well, yeah….. and IVI = V-I+I=V. Although there are semantic rules of cleaning and not going exceeding magnitudes in subtraction… such thinking is useful if you try to add/subtract/multiply roman numerals without decimal conversion[*].
And CXLVIII = 100+(50-10) + V+III = 148. What on earth is confusing about that?
[*] FOUR + ONE = FIVE because FOUR = (FIVE – ONE) and so (FIVE -ONE) + ONE = FIVE. The -ONE and +ONE cancel. IV +I = IVI, THe I’s before and after cancel. Other example: LXIV +XCVII = XCLXIVVII. The Xs one each side of CL cancel and single I on each side of VV cacel so = CLVVI. The VV combine to X and =CLXI. In other words 64 + 97 = (50+10 + (5-1)) + (100-10 + 5 + 2) = 100+50 + 5+5 + 1 = 100 + 50 + 10 +1 =161. Simple.
I dunno though. Dark Side of the Horse set it’s own level for sight puns that simply having 3 = square root of 9 and that’s a nerdy thing so we have a sheep in glasses, and a roman sheep leading a legion of arbitrarily 148 sheep just doesn’t seem to cut it. Unless there is some significance to 148.
Although there are semantic rules of cleaning and not going exceeding magnitudes in subtraction…
Woozy, yes, these are interesting to contemplate. I like some awkward examples that eschew even single subtractions, and will show four of a kind for some symbols. Going the other direction, it is disconcerting to see someone (mostly moderns) try to subtract across multi levels. The recent practical occasion for needing to try this out was of course the year 1999. Would you accept “IMM”? Or maybe “MIM”? Many stuck with “MCMXCIX”.
heres what I should have said.
VI is ONE after FIVE. So that is obviously SIX.
And IV is ONE before FIVE. So that is obviously FOUR.
You will occasionally see in (in tarot cards mostly) VIIII for nine, and occasionally you will see on clocks IIX for eight.
Once the nineties occurred and we sometimes saw MCMXCIII I always figured the Cs cancel and it should be MXMIII but I guess some people think substraction should only be one degree of magnitude. I get it but ….
Actually in the early 2000s I saw a date MCMCI and I figured someone after a century of MCM + (I to XCIX) just couldn’t break the habit.
Mitch, I think those of us using digits would thing 19 = 20 -1 would first think 1999 could be IMM but psuedo romans would think 19=(10 + 10) – 1 = 10 + (10-1) and do MIM. Of course real romans think in magitudes and an 1999 = 1000 + 900 + 90 + 9 so MCMXCIX but from a calculating point of view those cross cancelinng C’s and X’s are …. wow.
I agree that he’s obviously counting sheep to fall asleep, that he’s doing it in an odd way, and that Roman numbers can be confusing. But I assumed there was some significance to CXLVIII or 148. If the point was that Roman numbers can be confusing, it would make sense if IV was followed by IVI or something. But even someone confused about Roman numbers wouldn’t try to follow IV with CXLVIII.
Fun fact (which I’m sure most of you already know): on clock faces that use Roman numerals, at least on the old ones, 4 is represented as IIII.
I sometimes use Roman numerals to impress my students with just how easy math can be, and how much harder it can be. I usually also point out that the numbers they’re used to are Arabic numerals, because Arabians came up with lots of clever stuff.
@ Chak – The reason they do that is not for mathematical simplicity (nine is always “IX”), but for visual symmetry: the width of the “IIII” on the right harmonizes better with the “VIII” (eight) on the left.
BTW, I got the Bliss/Martin book! Hefty physical book. Now to go relax and read …
“the numbers they’re used to are Arabic numerals, because Arabians came up with lots of clever stuff.”
They did come up with lots of clever stuff. But much of what we label “Arabic” is only because it came to Europe via Arabia. Positional representation of numbers was invented by Hindus in China and/or India.
Reading Woozy’s explanation of Roman numerals brought to mind this quote from Johann Sebastian Bach: “There’s nothing remarkable about it. All one has to do is hit the right keys at the right time and the instrument plays itself.”
If you have a chance, visit the “History of Philosophy (without any gaps)” website / podcasts / books. A now-concluded sequence within the still-ongoing main Western thread was “Philosophy in the Islamic World” which was quite an eye-opener for me. https://historyofphilosophy.net/series/islamic-world
Becky/woozy: I started to write something about how Roman numeral rules are not straightforward or obvious, but then gave up, because how do you convince someone that rules they think are obvious are not obvious?
Mitchell and Webb’s Kitchen Nightmares @1:37:
Webb (Gordon Ramsey): “[Cooking this dish is easy.] It’s just local ingredients, simply cooked.”
Mitchell: “By you! King Lear is just English words put in order.”
I think I’ve mentioned this before, but I used to be a math major, and we did actually have conversations like this: “This step is trivial/obvious.” “I’m not sure, it is really obvious?” [Half an hour of consulting definitions, sketching proofs, etc. . .] “Oh yes, it is obvious, I see it now.”
If he jumps (!) from 4 to 148, it’s because the next 144 (=12*12) sheep come in a flock/legion instead of one by one, as is customary.
Now I can’t remember where, but I was going to bring up exactly this joke a couple days ago! Well, my version has the instructor / lecturer at the board say “This step is obvious … wait a minute ..” and check for several boards worth of space and most of the remaining time, to finally conclude “Yes, it is obvious”.
Oh now I remember! It was when everybody was yelling at me for calling the Frog Applause pun “straightforward”. I was ready to try to prove it! :-)
Interesting discussion, but I think dvandom had it at the first comment. I can relate to going to sleep peacefully and then the mind takes over.
“All one has to do is hit the right keys at the right time and the instrument plays itself.”
Rosenkrantz and Guildenstern would beg to differ:
Ham. I pray will you play vpon this pipe?
Ross. Alas my lord I cannot.
Ham. Pray will you.
Gil. I haue no skill my Lord.
Ham. Why looke, it is a thing of nothing,
T’is but stopping of these holes,
And with a little breath from your lips,
It will giue most delicate musick.
Gil. But this cannot wee do my Lord.
Ham. Pray now, pray hartily, I beseech you.
Ros. My lord wee cannot. (me?
Ham. Why how vnworthy a thing would you make of
You would seeme to know my stops, you would play vpon [G1]
You would search the very inward part of my hart, mee,
And diue into the secreet of my soule.
Zownds do you thinke I am easier to be pla’yd
On, then a pipe? call mee what Instrument
You will, though you can frett mee, yet you can not
Play vpon mee, besides, to be demanded by a spunge.
And I thought: Why does Shrug’s copy of Tom Stoppard have the dialogue in such old-fashioned spellings?
But roman numerals are easy.
To read them: I = 1; V=5, X=10; L=50;C=100;D=500; M=1000 and putting an overline increases a symbol by a magnitude of a thousand. They are list higher left to right by highest denominations first and you add them up. Because they are listed highest to lowest if you ever do see a lower denomination before a higher denomination that means the lower denomination is subtracted.
So CXLVIII = 100 + (50-10) + 5 + 3=148
That’s it!
To write them there some intuitive but hard, but not too hard, to formulize rules. If you can use larger denominations do so (Represent 150 and $CL and not LLL or CXXXXVV because the largest block you can use is C so you must use it). The primary units are I, X,C, M of magnitudes 1, 10,100,1000. V,L,D are “half units” magnitude 5x 1, 5x 10, 5x 100. Their purpose of existence is to allow us to represent five to nine units of a magnitude in fewer terms. To allow us even fewer terms placing a primary unit if the magnitude immediately less then the current primary or half unit, to the left of a unit (so we can think of it as subtraction) allows for us to express $4$ or $9$ of the lesser unit.
So 148 = 100 + 4×10 + 8×1 = 100 + (5×10-10) + 5×1 + 3x 1 = C + XL + VIII= CXLVIII.
And for me that’s the missing step!
If a legion is 12 x 12 rather than the 10x 10 as I assumed and if Horace is keeping a running tally (which hadn’t occurred to me but is clear in hind sight that he would be) then that explains the 148.
And that is enough to make a joke (whereas an arbitrary roman numeral would not).
But that’s gross!
Yes, I agree that if Roman legions came in groups of 12*12 that would make the joke make sense. But Olivier, is that actually the case, or were you just speculating that they might come in that size? I didn’t find any indication that that was a standard Roman legion size with a quick search.
According to a handy dictionary, a Roman legion was “A division of from 3000 to 6000 men (including cavalry) in the Roman army”.
Interesting article relating to Roman numerals and mathematics, and how the Roman numeral notations had changed.
https://washingtoncitypaper.com/article/191885/how-did-anyone-do-math-in-roman-numerals/
I don’t think the “148” that Samson picked has any “meaning”: it was probably just selected to make the number sufficiently complex, without being unreasonably large. As noted above, it is far too small for a “legion”, but also too large for the “century” that a “centurion” would command. The size of these “platoons” varied considerably (from 30 to 80, depending on the era), but never reached 100, let alone 144 soldiers.
P.S. @ WW – Back in college, one alleged strategy for solving math problems was to work down from the top equation, then work up from what the solution was known to be, and then write in the middle (between the two parts): “…it is thus intuitively obvious that…”
P.P.S. One guy who actually tried this trick got a note back from the T.A. that said “It is to me, but I don’t think it was to you.” (with points deducted).
Still trying to figure out if 148 has any significance here, or if it’s just a number that’s comically complicated to express in Roman numerals. I like the “144 more” idea but I don’t know if that’s a specific Roman unit size either.
Re: clocks with IIII for 4: I was always told that it was a superstitious avoidance of the letters “IV” which could refer to IVPPITER. Seems to be an urban legend, since the “subtractive” notation appears to be a medieval innovation that didn’t come along until centuries after anyone cared about Jupiter except perhaps as a planet.
https://www.straightdope.com/21341605/why-do-clocks-with-roman-numerals-use-iiii-instead-of-iv
A riddle my uncle used to like:
A dozen is twelve.
A baker’s dozen is thirteen; a regular dozen with one more.
A gross is 144; a dozen dozens.
So what is a baker’s gross?
Answer: chicken guts in the doughnuts.
Kilby: I suspect that explicitly writing “it’s intuitively obvious,” or “it trivially follows” is actually more likely to call the T.A.’s attention to that line, and make them question if you know what you’re doing, than just saying “therefore” or “it follows.”
Yes, and I think we’re onto something with these thoughts about “a gross”.
All right, he’s asleep or hypnagogic, and logic is reduced. So the appearance of Roman soldiers ties into the Roman numerals, but doesn’t oblige the narrative to make them a legion or a century or anything Roman. Instead, Horace reverts to his sometimes business persona, and puts in an order for a common commercial measure of additional sheep, one gross. They are arriving in one bunch, and his running total reaches the number shown.
There we are, Cello! Sorry, I meant Viola! Sorry, I meant Wah-La!
On “Obvious” — At one time I worked for the Post Office. We trainees were given overviews of some matters we might or might not be dealing with on a daily basis, including various regulations. One set of rules was about how the station and individual carriers and mail handlers needed to treat various kinds of mail. Some important rules were about First Class mail of course. A different set protected Third, or Parcels, “of obvious value”.
Mailrooms and shippers knew all about postal regs. To make sure their Third Class and Parcel mail was handled safely and promptly, they wanted to ensure these pieces would count as being “of obvious value”. Some of them had a big rubber stamp (standard issue from stationery stores) and marked their mail .. wait for it … no not “VALUABLE” which you’d think would make it obvious … but instead “OBVIOUS VALUE”.
So this discussion has made me realize that you can mathematically prove that an MD is worth more than a JD:
MD = 1500
JD = ID = 499
QED
Since we’re 41 comments in with no clear explanation of why 148 specifically was chosen, I’m guessing that it’s just a randomly chosen number. If it was another comic, I would have accepted that much sooner, but with Dark Side of the Horse I have more of an expectation of an “Aha!” insight after a potentially very long wait.
So you don’t accept the “got to IV by single counting, then ordered another gross from the sheep supply — 4 + 144 = 148 = CXLVIII”?
Oh, sorry, I actually thought you were joking! :o
That seems pretty convoluted to me. There’s no sign that he’s resorting to a business persona, and why would the 144 sheep that he ordered be carrying a Roman banner (and wearing uniforms, I think)? I think it’s more likely that this is just a bunch of Roman sheep following their leader, and that 144 is just a randomly chosen large number.
” If it was another comic, I would have accepted that much sooner, but with Dark Side of the Horse I have more of an expectation of an “Aha!” insight after a potentially very long wait.”
Exactly!
“Oh, sorry, I actually thought you were joking!”
Me too but I thought he was joking about stuffing donuts down his pants too.
larK says:
So this discussion has made me realize that you can mathematically prove that an MD is worth more than a JD:MD = 1500JD = ID = 499But with that prompt we cannot forget to remind everyone of the famous point that Halloween and Christmas are just notational variants:
31 OCT == 25 DECThe problem with Roman legions is that their composition has varied throughout history.
Roughly, one legion is made of 10 cohorts, each cohort holds 3 or 4 maniples comprising 120 to 160 men. One maniple is 2 centuries.
In the Straight Dope article, they mentioned sundials. Indeed, examples of ancient Roman sundials did use IIII so it’s possible that early clock makers copied that style.
Following from Olivier’s hint:
For IV we have one centurion. That might be followed by two centurions. Instead, we have 148. A common size of a maniple (two centuries of 80) is 160. In the earlier panels, we have 1 + 2 + 9 = 12, which combined with 148 makes a those two centuries (instead of two centurions).
Q.E.D. Simple and straightforward :)
A more interesting question to ask would be whether there is some significance to 148 in Finland, the cartoonist’s home country.
Q.E.D. Simple and straightforward
Intuitively obvious to the most casual observer. And they don’t get much more casual than this observer, amiright people?
It’s a nice theory, except that in the earlier panels, we have 1+2+√9=6, and not 12.
Ah, but the panels are square, and there are three of them, so you get 6^2/3… 12!
I’ve been trying to find a modern military unit of 144 men ; the range would make it a company, but there are huge variations here as well.
Actually, if you want 144, you can buy plastic toy soldiers: it seems they’re sold by the gross.
@ Mitch4 – “Halloween and Christmas are just notational variants:
31 OCT == 25 DEC”
Does that mean that the website subheader for next month will read
“It’s OctEmber!” ?
P.S. After all (as Churchill LaFemme once observed), Christmas falls on October 86th.
And as I think I’ve mused here before, “740” used to be The Number of the Beast, before the Beast’s agent took our his ten per cent commission.
Could this be a reference to Caesar’s 5th legion?
Mitch4 –
So, is the fact that my mail is not stamped “obvious value” why instead of forwarding (from our PO Box) some of it they are returning some of it to the sender (NYS Tax and a credit card company) or it just disappears (16 bank statements for us, our business, and the two clubs of which I am treasurer)in 3 months) – and then when we got desperate enough to make a late night trip to the PO mid corona virus stay at home – there was mail (first class again) sitting in our PO box?
Worst of all we were having our mail go to our PO box – at a post office other than the one which delivers to our house – because we had been having a problem with mail going astray and not getting to our house, so other than the returned mail and what was in the box when we went, we have no idea of which post office (our home post office or our PO box post office) is the one losing our mail. I am currently waiting for 2 checks to come from various places which are long over do since mailing plus another check had to be replaced as it never made it here.
I so much miss and appreciate the daily trips to our PO Box for mail in and mail going out.
Recently read an article in BBC History magazine that explained Roman Numerals were taken from the Etruscan system, They used different symbols which were more complicated, but they receive no credit for their system being used to develop the Roman system.
Ay ay ay, who knows what is up with the mail these days! But my experience with the “obvious value” stamp was from the summer of Apollo 11 and Woodstock, so things may have changed by now, as they say on the NPR politics round-up podcast.
@MArk H.: how so?
“but they receive no credit for their system being used to develop the Roman system.”
Well, considering the general impression is that the Roman numbers are unyieldy and a poor choice for calculations (I’ve only been arguing Roman numerals are straightfoward and not that bad; I’m not arguing that they have any advantage) I don’t think many people think that there is much credit deserved.
(After all, if they hadn’t used Etruscan numbers they’d have used something else…..)
(Actually, just how many possible number systems are possible, really?)
@ woozy – How many? I’m not sure whether I’m up for a discussion about all the multitudinous varieties (meaning magnitudes) of “infinity”. I’ve got one kid who has gotten as far as “fractions” in school (just adding and subtracting, they recently started multiplying them, but division is a long way off). He’s aware of negative numbers, but he picked that up from me, they haven’t touched them in school yet. I’ve already warned him that he will have to get used to several more massive expansions to the “set of known numbers” (all he has right now are natural numbers and a dusting of rationals, there’s still integer, real, irrational, and complex numbers, and probably several other types that Winter Wallaby could add).
And just wait until you get mugged by the ordinal infinites. So much more structure than the cardinals have, even with their surprises.
One of my students was taking a course to prepare her to teach math to k-8 children (dog help those kids, but that’s another story) and she was learning (sorta) about various historical number systems. The one I particularly liked was base 60. It was fun, for me at least.
It was fun for a minute. Or a second-minute.
Well, just to a degree.
@ Chak – I took a “History of Math” class in college that did a similar sort of review. At least for the Babylonian stuff all we had to do was learn the mathematical notation, the professor didn’t expect us to learn any matching vocabulary (like he did for Greek).
P.S. @ Danny Boy – LºL ;-)
Speaking of modular or Base arithmetic, one of the oddest but still workable ones I’ve seen is what I recall as “Base-3 subtractive” but this Wikipedia article titles “Balanced Ternary”. Like “normal” ternary, it has three digits, but instead of {0,1,2} they are {-1,0,1}. (In the article, and generally, a letter is used for -1, here T.) As their chart shows, you can represent and count to numbers in a predictable and regular way, though maybe hard to do in your head.
The article also shows how to do … not “decimals” I guess but fractionals written with digits to the right of a radix point.
I was surprised at how old the article says the system is. I thought I must have learned of it from something like a Martin Gardner column, and that it was credited to somebody like Conway, and was especially suited for representing the solutions or strategy of some puzzle or game.
Horace never gives up on sheep-counting. Even when reduced to more-usual numerals.
@ Mitch4 – I thought that the name “Conway” sounded familiar, but only after I looked him up did I remember that he invented the “game of life”. I’m sure there are tri-state variations of the game in which the cells might be described as “live”, “dead”, and “sick” (or some other amusing adjective).
Conway was on my mind after recently hearing the special memorial edition of BBC More or Less, linked here . The posted summary: “Mathematician John Horton Conway died in April this year from complications related to Covid-19. We remember the man and his work.” Some of it is about him ruefully accepting always having Cellular Automata (Game of Life) mentioned first among his achievements.
For reading, the Wikipedia article on him is quite interesting.
Their lead summary paragraph:
John Horton Conway FRS (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life.I think Stephen Wolfram would object to cellular automata being dismissed as “recreational mathematics”…
Ditto John Conway.
I wonder if they would really consider that dismissive. I’m not a mathematician (although as a physicist I did research in some cellular automata models) and for me the “recreational” adjective means that it’s particularly awesome, rather than excluding the possibility of usefulness.
Right, in the Wikipedia article take note of the section on Conway’s friendship with Martin Gardner, the longtime “Mathematical Games” columnist at Scientific American magazine and a key figure in the popularity of recreational math as a recognized hobby or amusement , attractive to many nerdy ordinary people but also to serious mathematicians.
Mitch4 wrote: “Some of it is about him ruefully accepting always having Cellular Automata (Game of Life) mentioned first among his achievements,” and the Wikipedia article kind of dismisses it as an additional bit of trivia; meanwhile, Steven Wolfram puts cellular automata front and center in his New Kind of Science. Now, it’s not clear whether Wolfram is really on to something or not, but if he is, then cellular automata just might be the most important thing Conway ever did…
I appreciate that real math geeks don’t see “recreational” as a disparagement, but then they have interesting takes on common words (eg: trivial) that most people wouldn’t properly appreciate…
The true math geek in me finds the word “recreational” redundant. What field of math isn’t recreational?
I worked my way thru much of Winning Ways and enjoyed it very much, even when getting beyond my depth with the surreal numbers. It’s also a lovely physical book, with even some color printing so diagrams can easily distinguish stuff by putting things in red and blue. If you can get hold of it, I bet you would enjoy it too.
The mentions of Conway working on “the theory of games” is not the same thing as “game theory” in the sense of von Neumann & Morgenstern and then “Nash equilibrium” and all that stuff. No, it’s games like Nim. (These are the subject of Winning Ways and On Numbers and Games .)
Do you know Nim? Shall we play a couple rounds? Warning: it is a solved game. There are winning positions and losing positions.
You put down on the table several rows of some sort of token. Players take turns, on each turn removing one or more tokens from a single row of the layout. You cannot “pass” and take away none. You cannot take away from more than one row. The player who is able to take the last remaining token and clear the table is the winner. (It is often played the opposite way — the player “forced” to take the last token loses. The analysis is the same, in that “safe” and “losing” positions along the way are actually the same, up until you have to switch at the end.)
Let’s use the classic “Marienbad” layout, of 1-3-5-7.
1 *3 * * *5 * * * * *7 * * * * * * *This is a winning position (to leave). So to be fair, I will go first, taking some away, and leaving a losing position (to leave) — which your move can convert to a winning position.
I will take 2 from the row of 7:
1 *3 * * *5 * * * * *5 * * * * *Now it’s up to you …
To clarify the relationship of winning and losing positions:
— If one player leaves a winning position, every possible move by the opponent leaves a losing position
— If one player leaves a losing position, there exists at least one move by the opponent which leaves a winning position
The “solution” for winning play is a formula or algorithm for finding the move, presented with a losing position, which will produce a winning position. Once you have left a winning position, if you use the formula or algorithm, you can continue to receive losing positions from the opponent and convert them to winning positions.
Mitch4: I play the Calvinball gambit, of changing the rules absurdly to something like “But then the universe declares the tallest player wins.” Or “suddenly Hulk Hogan appears in a puff of smoke and says that the player with the longest muttonchops wins, Brotha.” Looks good for me.
Your turn.
Mitch4: I loved that book. The analysis of how to play Dots-And-Boxes was amazing.
Shrug: A strange game. The only winning move is not to play.
For a nice take on the use of ordinary words for special ideas in math, look up ‘Finite simple group of order two’ by the Klein Four group on YouTube. I’d give a URL but I can’t right now.
@ Chak – I found the link to the “Finite Simple Group of Order Two”, but I think most of the humor will be lost upon non-math majors.
@ WW – I was quite amused by Shrug’s permutation, precisely because it led to your “winning” move. Even though Mitch4 gave a very precise description of the game, I cannot do the binary arithmetic for the four row layout in my head (I know all the winning patterns for the 3-4-5 layout, so that one doesn’t require any “work” for me). In addition, I vastly prefer the “last move loses” version.
You don’t have to do it in your head, you can do it on your fingers! Right pinky is ones, right ring is twos, right middle is fours, right index is eights …
To evaluate the original 1 3 5 7, the 1 gives you pinky down.
The 3 reverses the twos and the ones, so it’s ring down and pinky up.
The 5 is the middle and pinky down, leaving last three fingers down.
The 7 is last three fingers, so they come back up.
Leaving no fingers down, so it is a safe position, and any move leaves unsafe.
For my leave of 1 3 5 5 you could also start with the fingers, But easier to note that the two 5 5 cancel , so we have 3 and 1, which reduce to a 2. The only 2 in that board is in the 3 (not in the 5s!) so we will take 2 from the 3, leaving
1 *
1 *
5 * * * * *
5 * * * * *
which, almost by inspection is a winning position to leave. We will just match whatever the opponent does, in matching row.
It is not yet at the point where strategy changes for Kilby’s preferred misêre version, as they are called.
“I vastly prefer the “last move loses” version”
But that makes absolutely no difference in evaluating this position, or almost any position well short of the endgame.
I don’t carry the 3 4 5 positions in my memory, but it is quick enough on fingers or simply mentally to calculate the Nim Value (as they call it) to be 2. So take 2 from the 3, leaving 1 4 5. And again, just as we could see the pair of 5 5 cancelling each other in the other example, this is only slightly harder to see as 1 plus even n plus odd n+1, a common 0 grouping.
And even though small, this is still not quite endgame, and makes no difference between normal play and misère version.
@ Mitch4 – Cool, I like the “binary finger arithmetic” algorithm. I’ll try to teach that trick to my son.
Thanks for the link, Kilby.
Cool, I like the “binary finger arithmetic”
Back when my joints and muscles had more agility, and I more often found myself in waiting-around situations without amusements, I would transform the “tapping your fingers” nervous gesture into binary counting. I maintained I could reach 1024 error-free but I was probably kidding myself.
I used to know all sorts of math – but being an accountant – the basics of arithmetic, estimating, and trying to figure out what the client’s numbers mean is about all I deal with these days. I used to be much better at doing arithmetic in my head, but since I started using a calculator and then a computer much of it has left my head so I have room for less important information – such as why my colonial self and her mother will never actually “own” the family property, just get to use it for life with my nephew ending up owning it after we are gone, etc. These days I am glad if I can figure out what day it is without looking at my day of the week pill box.
I suppose he qualifies, under the meaning of “typical male citizen relaxing at home”. But he doesn’t seem to actually have vi of anything going on, just that i cup, and i jug.