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.

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 arethreeof 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 DECDoes 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

ordinalinfinites. So much more structure than thecardinalshave, 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 Waysand 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

notthe 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 ofWinning WaysandOn 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

winningposition (to leave). So to be fair, I will go first, taking some away, and leaving alosingposition (to leave) — which your move can convert to awinningposition.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.