# How to count

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Let me tell you about a new game that we’ve been playing in our family, the rule-making game. It is a talking game, requiring no pieces or objects of any kind, and it can easily be played whilst walking or traveling. My children and I recently played several rounds of it walking around London on a recent visit there.

The game has no rules, initially, nor even any definite procedure — it is different every time — but things usually become clear soon enough. It usually makes a better game to cooperate on the first several turns to lay the groundwork.

Let me explain how to play simply by example:

**Papa**: The first rule is that the players shall take turns making rules, and that every rule shall have a rule number, which is incremented on each turn.

**Horatio**: The second rule is that the players must state their rules in the form, “The first rule is…” or “the second rule is…” and so on, and that players are not allowed to ask what is the current rule number, or they lose.

**Hypatia**: The third rule is that the other players must say, “thank you” after another player makes a rule.

(… “thank you”…. “thank you”….)

**Papa**: The fourth rule is that the rules must not contradict each other, and no rule is allowed that abrogates an earlier rule.

(… “thank you”…. “thank you”….)

**Horatio**: The fifth rule is that after making an odd-numbered rule, the player must stomp on the ground.

(STOMP… “thank you”…. “thank you”….)

**Hypatia**: The sixth rule is that no player may win immediately after their own rule.

(… “thank you”…. “thank you”….)

**Papa**: The seventh rule is that right after a player stomps according to rule five, the other two players must hop.

(STOMP … “thank you”…. “thank you”….HOP….HOP…)

**Horatio**: The eighth rule is that if a player loses, then the game continues without that person.

(… “thank you”…. “thank you”….)

**Hypatia**: The ninth rule is that after stating a rule, the other two players must state a different color.

(STOMP … “thank you”…. “thank you”….HOP…HOP… “blue”… “green”…)

**Papa**: The tenth rule is that furthermore, those colors must never repeat, and they must be stated simultaneously, on the count of 1-2-3.

(… “thank you”…. “thank you”…. “1-2-3: neon green / violet”)

**Horatio**: The eleventh rule is that if there is only one player left, then that player wins.

(STOMP … “thank you”…. “thank you”….HOP…HOP… “1-2-3: red/orange”)

**Hypatia**: The twelfth rule is that every player must jump up and down (…jump…) while stating their rule. (….jump jump jump…)

(… “thank you”…. “thank you”…. “1-2-3: pink/turquoise”)

**Papa**: (jump jump…) The thirteenth rule is that (…jump…) in the case of dispute (…jump…), the question of whether or not someone has violated or followed a rule shall be decided by majority vote (…jump…).

(STOMP … “thank you”…. “thank you”….HOP…HOP… “1-2-3: yellow/brown”)

**Horatio**: (jump….) The fourteenth rule is that (…jump…) before stating their rule, the players must state a country, and that whoever repeats a country loses (…jump…)

(… “thank you”…. “thank you”…. “1-2-3: black/gray”)

**Hypatia**: (jump…) Germany. The fifteenth rule is that (…jump…) there can be at most twenty-five rules.

(STOMP … “thank you”…. “thank you”….HOP…HOP… “1-2-3: sky blue / peach”)

**Papa**: (jump…) United States. The sixteenth rule is that (…jump…) if all current players lose at the same time after a rule, then the player previous to that rule-maker is declared the “honorary winner”. (…jump…)

(… “thank you”…. “thank you”…. “1-2-3: white / white”)

Oh no! Since both Horatio and Hypatia said “white”, they both lose. And then Papa also loses in light of rule six. So we’ve all lost! But then, in light of rule sixteen, Hypatia is declared the honorary winner! Hooray for Hypatia!

I hope you all get the idea. Please enjoy! And report your crazy or interesting rules in the comments below.

As a guest today in my daughter’s second-grade classroom, full of math-enthusiastic seven-and-eight-year-old girls, I led a mathematical investigation of graph coloring, chromatic numbers, map coloring and Eulerian paths and circuits. I brought in a pile of sample graphs I had prepared, and all the girls made up their own graphs and maps to challenge each other. By the end each child had compiled a mathematical “coloring book” containing the results of their explorations. Let me tell you a little about what we did.

We began with vertex coloring, where one colors the vertices of a graph in such a way that adjacent vertices get different colors. We started with some easy examples, and then moved on to more complicated graphs, which they attacked.

The aim is to use the fewest number of colors, and the chromatic number of a graph is the smallest number of colors that suffice for a coloring. The girls colored the graphs, and indicated the number of colors they used, and we talked as a group in several instances about why one needed to use that many colors.

Next, the girls paired off, each making a challenge graph for her partner, who colored it, and vice versa.

Map coloring, where one colors the countries on a map in such a way that adjacent countries get different colors, is of course closely related to graph coloring.

The girls made their own maps to challenge each other, and then undertook to color those maps. We discussed the remarkable fact that four colors suffice to color any map.

Next, we considered Eulerian paths and circuits, where one traces through all the edges of a graph without lifting one’s pencil and without retracing any edge more than once. We started off with some easy examples, but then considered more difficult cases.

An Eulerian *circuit* starts and ends at the same vertex, but an Eulerian *path* can start and end at different vertices.

We discussed the fact that some graphs have no Eulerian path or circuit. If there is a circuit, then every time you enter a vertex, you leave it on a fresh edge; and so there must be an even number of edges at each vertex. With an Eulerian path, the starting and ending vertices (if distinct) will have odd degree, while all the other vertices will have even degree.

It is a remarkable fact that amongst connected finite graphs, those necessary conditions are also sufficient. One can prove this by building up an Eulerian path or circuit (starting and ending at the two odd-degree nodes, if there are such); every time one enters a new vertex, there will be an edge to leave on, and so one will not get stuck. If some edges are missed, simply insert suitable detours to pick them up, and again it will all match up into a single path or circuit as desired. (But we didn’t dwell much on this proof in the second-grade class.)

Meanwhile, this was an excellent opportunity to talk about The Seven Bridges of Königsberg. Is it possible to tour the city, while crossing each bridge exactly once?

The final result: a booklet of fun graph problems!

The high point of the day occurred in the midst of our graph-coloring activity when one little girl came up to me and said, “I want to be a mathematician!” What a delight!

Andrej Bauer has helped me to make available a kit of my original uncolored pages (see also post at Google+), if anyone should want to use them to make their own booklets. Just print them out, copy double-sided (with correct orientation), and fold the pages to assemble into a booklet; a few staples can serve as a binding.

See also my followup question on MathOverflow, for the grown-ups, concerning the computational difficulty of producing hard-to-color maps and graphs.

Finally, check out my new book:

*Doubled, squared, cubed* is a great math game to play with kids or anyone interested in math. It is a talking game, requiring no pieces or physical objects, played by a group of two or more people at almost any level of mathematical difficulty, while sitting, walking, boating or whatever. We play it in our family (two kids, ages 7 and 11) when we are sitting around a table or when walking somewhere or when traveling by train. I fondly recall playing the game with my brothers and sisters in my own childhood.

The game proceeds by first agreeing on an allowed number range. For youngsters, perhaps one wants to allow the integers from 0 to 100, inclusive, but one will want to have negative numbers soon enough, and of course much more sophisticated play is possible. Eventually, one lessens or even abandons the restriction altogether. The first player offers a number, and each subsequent player in turn offers a mathematical operation, which is to be applied to the current number, which must not be mentioned explicitly. The resulting number must be in the allowed number range.

The goal of the game is successfully to keep track of the number as it changes, and to offer an operation that makes sense with that number, while staying within the range of allowed numbers. The point is to have some style, to offer an operation that proves that you know what the number is, without stating the number explicitly. Perhaps your operation makes the new number a nice round number, or perhaps your operation can seldom be legally applied, and so applying it indicates that you know it is allowed to do so. You must offer only operations that you yourself can compute, and which do not rely on hidden information (for example, “*times the number of grapes I ate at breakfast*” is not really permissible).

A losing move is one that doesn’t make sense or that results in a number outside the allowed range. In this case, the game can continue without that person, and the last person left wins. It is not allowed to offer an operation that can always be applied, such as “*times zero”* or “*minus itself*“, or which can always be applied immediately after the previous operation, such as saying “*times two” *right after someone said, “*cut in half”.* But in truth, the main point is to have some fun, rather than to win. Part of the game is surely simply to talk about new mathematical operations, and we usually take time out to discuss or explain any mathematical issue that may come up. So this is an enjoyable way for the kids to encounter new mathematical ideas.

Let me simply illustrate a typical progression of the game, as it might be played in my family:

**Hypatia:** *one*

**Barbara:** *doubled*

**Horatio:** *squared*

**Joel:** *cubed*

**Hypatia:** *plus 36*

**Barbara:** *square root*

**Horatio:** *divided by 5*

**Joel:** *times 50*

**Hypatia:** *minus 100*

**Barbara:** *times 6 billion*

**Horatio:** *plus 99*

**Joel:** *divided by 11*

**Hypatia:*** plus 1*

**Barbara:** *to the power of two*

**Horatio:** *minus 99*

**Joel:** *times itself 6 billion times*

**Hypatia:** *minus one*

**Barbara:** *divided by ten thousand*

**Horatio:** *plus 50*

**Joel:** *plus half of itself*

**Hypatia:** *plus 25*

**Barbara:** *minus 99*

**Horatio:** *cube root*

**Joel:** *next prime number above*

**Hypatia:** *ten’s complement*

**Barbara:** *second square number above*

**Horatio:** *reverse the digits*

**Joel:** *plus 3 more than six squared*

**Hypatia:** *minus 100*

and so on!

As the kids get older, we gradually incorporate more sophisticated elements into the game, and take a little time out to explain to young Hypatia, for example, what it means to cube a number, to take a number to the power two, or what a prime number is. I remember playing the game with my math-savvy siblings when I was a kid, and the running number was sometimes something like $\sqrt{29}$ or $2+3i$, and a correspondingly full range of numbers and operations. It is fine to let the youngest drop out after a while, and continue with the older kids with more sophisticated operations; the youngsters will rejoin in the next round. In my childhood, we had a “challenge” rule, used when someone suspects that someone else doesn’t know the number: when challenged, the person should say the number; if incorrect, they are out, and otherwise the challenger is out.

Last weekend, I played the game with Horatio and Hypatia as we walked through Central Park to the Natural History Museum, and they conspired in whispering tones to mess me up, until finally I lost track of the number and they won…

My recent talk, Playful Paradox with large numbers, infinity and logic, was a romp through various paradoxical topics in mathematics and logic for an audience of about 150 mathematics and philosophy undergraduate students at Fudan University in Shanghai, and since Fudan is reportedly a top-three university in China, you can expect that the audience was sharp.

For a bit of fun, and in keeping with the theme, we decided to hold a **Largest-Number Contest**: what is the largest number that one can describe on an ordinary index card?

My host Ruizhi Yang made and distributed announcement flyers for the contest a week before the talk, with a poster of the talk on one side, and a description of the contest rules on the other, and including a small card to be used for submissions. The rules were as follows:

- A submission entry consists of the description of a positive integer written on an ordinary index card, using common mathematical operations and notation or English words and phrases.
- Write legibly, and use at most 100 symbols from the ordinary ASCII character set. Bring your submission to the talk.
- Descriptions that fail to describe a number are disqualified.
- The submission with the largest number wins.
- The prize will be $1 million USD, divided by the winning number itself, rounded to the nearest cent, plus another small token prize.

Example submissions:

$99999$.

$10*(10*99)+5$

*The population of Shanghai at this moment.*

The submissions were collected at the beginning of my talk, which I began with a nod to Douglas Hofstadter, who once held a similar contest in his column at Scientific American magazine.

To win, I began, it seemed clear that the submitted number must be very large, and since the prize was to be one million dollars *divided* by the winning number, then this would mean that the prize should become small. So the goal is just to win, and not necessarily to win any money. The real prize is Pride.

So, what kind of numbers should one submit? Of course, anyone could think to write $99999$, or one million ($10^6$), one billion ($10^9$) or one trillion ($10^{12}$). Perhaps someone would want to write a *googol*, which is a common name for $10^{100}$. Written in decimal, this is $$10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,$$ which is one character too long for the rules, so one should use the more compact formulations. But let’s just write it as *googol*.

But we can easily describe much larger numbers. A *googol plex* is the number $10^{\text{googol}}$, written in decimal with a $1$ followed by a *googol* many $0$s. A *googol plex* is also describable as $10^{10^{100}}$. An iterated exponential $a^{b^c}$ could be seen as ambiguous, referring either to $(a^b)^c$ or to $a^{(b^c)}$. These are not always the same, and so exponentiation is not an associative operation. But which is bigger? If the aim is to make big numbers, then generally one wants to have as big an exponent as possible, and so $a^{(b^c)}$ will be much larger than $(a^b)^c$, which is just $a^{bc}$, and so we will always associate our exponentials upward. (Nevertheless, the upward associated form is not always larger, which you can see in the case of $2^{1^{100}}$.) In general, we can apply the *plex* operation to any number, so $x$ *plex* just means $10^x$.

A *googol bang* means *googol *!, or *googol* factorial, the number obtained by multiplying *googol* $\cdot$ (*googol* -1) $\cdots 2\cdot 1$, and in general, $x$ *bang* means $x!$. Which is bigger, a *googol plex* or a *googol bang*? If you think about it, a *googol plex* is 10 times 10 times 10, and so on, with a *googol* number of terms. Meanwhile, the factorial *googol bang* also has a *googol* terms in it, but most of them (except for 10) are far larger than 10, so it is clear that a *googol bang* is much larger than a *googol plex*.

Continuing down this road, how about a *googol plex bang*, or a *googol bang plex*? Which of these is larger? Please turn away and think about it for a bit……..Have you got it? I think like this: A *googol plex bang* has the form $(10^g)!$, which is smaller than $(10^g)^{10^g}$, which is equal to $10^{g10^g}$, and this is smaller than $10^{g!}$ when $g$ is large. So a *googol plex bang* is smaller than a *googol bang plex*.

There is an entire hierarchy of such numbers, such as *googol bang plex bang bang bang* and *googol bang bang plex plex plex*. Which of these is bigger? Do you have a general algorithm for determining which of two such expressions is larger?

A *googol stack* is the number

$$10^{10^{10^{\cdot^{\cdot^{10}}}}}{\large\rbrace} \text{ a stack of height googol},$$

where the height of the exponential stack is *googol*. Similarly, we may speak of $x$ *stack* for any $x$, and form such expressions as:

*googol stack bang plex*

*googol bang plex stack*

*googol plex stack bang*

Which of these is larger? If you think about it, you will see that the* stack* operator is far more powerful than *bang *or *plex*, and so we want to apply it to the biggest possible argument. So the second number is largest, and then the third and then the first, simply because the *stack* operator will overrule the other effects. Let us call this hierarchy, of all expressions formed by starting with *googol* and then appending any number of *plex, bang* or *stack* operators, as the googol plex bang stack hierarchy.

**Question.** Is there a feasible computational procedure to determine, of any two expressions in the *googol stack bang plex* hierarchy, which is larger?

This is an open question in the general case. I had asked this question at math.stackexchange, Help me put these enormous numbers in order, and the best currently known (due to user Deedlit) is that we have a simple algorithm that works for all expressions of length at most *googol*-2, and also works for all expressions having the same number of operators. But this algorithm breaks down for large expressions of much different lengths, since it seems that one must come to know the exact threshold where an enormous iteration of *bang*s and *plex*es can ultimately capture a *stack*. Of course, in principle one can compute these numbers exactly, given sufficient time and space resources, but the point of the question is to have a *feasible* algorithm, such as one that takes only polynomial time in the length of the input expressions. And a feasible such general algorithm is not known.

The fact that we seem to have difficulty comparing the relative sizes of expressions even in the simple *googol plex bang stack* hierarchy suggests that we may find it difficult to select a winning entry in the largest-number contest. And indeed, I’ll explain a bit later why there is no computable procedure to determine the winner of arbitrary size largest-number contests.

The numbers we’ve seen so far are actually rather small, so let’s begin to describe some much larger numbers. The most basic operation on natural numbers is the successor operation $a\mapsto a+1$, and one can view addition as repeated succession:

$$a+b=a+\overbrace{1+1+\cdots+1}^{b\text{ times}}.$$

Similarly, multiplication is repeated addition

$$a\cdot b=\overbrace{a+a+\cdots+a}^{b\text{ times}}$$

and exponentiation is repeated multiplication

$$a^b=\overbrace{a\cdot a\cdots a}^{b\text{ times}}.$$

Similarly, stacks are repeated exponentiation. To simplify the expressions, Knuth introduced his uparrow notation, writing $a\uparrow b$ for $a^b$, and then repeating it as

$$a\uparrow\uparrow b=\overbrace{a\uparrow a\uparrow\cdots\uparrow a}^{b\text{ times}}= a^{a^{a^{.^{.^a}}}}{\large\rbrace}{ b\text{ times}}.$$

So *googol stack* is the same as $10\uparrow\uparrow$ *googol*. But why stop here, we can just as easily define the triple repeated uparrow

$$a\uparrow\uparrow\uparrow b=\overbrace{a\uparrow\uparrow a\uparrow\uparrow\ \cdots\ \uparrow\uparrow a}^{b\text{ times}}.$$

In this notation, $10\uparrow\uparrow\uparrow$ *googol* means

$$\overbrace{10^{10^{.^{.^{10}}}}{\large\rbrace}10^{10^{.^{.^{10}}}}{\large\rbrace}10^{10^{.^{.^{10}}}}{\large\rbrace}\cdots\cdots{\large\rbrace} 10^{10^{.^{.^{10}}}}{\large\rbrace}10}^{\text{iterated googol times}},$$

where the number of times the stacking process is iterated is *googol*. This number is a vast number, whose description $10\uparrow\uparrow\uparrow$ *googol* fits easily on an index card.

But why stop there? We could of course continue with $10\uparrow\uparrow\uparrow\uparrow$ *googol* and so on, but I’d rather define

$$10\Uparrow x=10\ \overbrace{\uparrow\uparrow\cdots\uparrow}^{x\text{ times}}\ 10.$$

So that $10\Uparrow$ *googol* is vaster than anything we’ve mentioned above. But similarly, we will have $10\Uparrow\Uparrow$ *googol* and so on, leading eventually to a triple uparrow

$$10 \uparrow\hskip-.55em\Uparrow\text{ googol } = 10 \overbrace{\Uparrow\Uparrow\cdots\Uparrow}^{\text{googol iterations}} 10,$$

and so on. These ideas lead ultimately to the main line of the Ackerman function.

So we’ve described some truly vast numbers. But what is the winning entry? After all, there are only finitely many possible things to write on an index card in accordance with the limitation on the number of symbols. Many of these possible entries do not actually describe a number–perhaps someone might submit a piece of poetry or confess their secret love–but nevertheless among the entries that do describe numbers, there are only finitely many. Thus, it would seem that there must be a largest one. Thus, since the rules expressly allow descriptions of numbers in English, it would seem that we have found our winning entry:

*The largest number describable on an index card according to **the rules of this contest*

which uses exactly 86 symbols. We seem to have argued that this expression describes a particular number, and accords with the rules, so we conclude that this is indeed provably the winning entry.

But wait a minute! If the expression above describes a number, then it would seem that the following expression also describes a number:

*The largest number describable on an index card according to **the rules of this contest, plus one*

which uses 96 symbols. And this number would seem to be larger, by exactly one, of the number which we had previously argued must be the largest number that is possible to describe on an index card. Indeed, how are we able to write this description at all? We seem to have described a number, while following certain rules, that is strictly larger than any number which it is possible to describe while following those rules. Contradiction! This conundrum is known as Berry’s paradox, the paradox of “the smallest positive integer not definable in under eleven words.”

Thus, we can see that this formulation of the largest-number contest is really about the Berry paradox, which I shall leave you to ponder further on your own.

I concluded this part of my talk by discussing the fact that even when one doesn’t allow natural language descriptions, but stays close to a formal language, such as the formal language of arithmetic, then the contest ultimately wades into some difficult waters of mathematical logic. It is a mathematical theorem that there is no computable procedure that will correctly determine which of two descriptions, expressed in that formal language, describes the larger natural number. For example, even if we allow descriptions of the form, “the size of the smallest integer solution to this integer polynomial equation $p(x_1,\ldots,x_k)=0$,” where $p$ is provided as a specific integer polynomial in several variables. The reason is that the halting problem of computability theory is undecidable, and it is reducible to the problem of determining whether such a polynomial equation has a solution, and when it does, the size of the smallest solution is related to the halting time of the given instance of the halting problem. Worse, the independence phenomenon shows that not only is the comparison problem undecidable, but actually there can be specific instances of descriptions of numbers in these formal languages, such that both definitely describe a specific number, but the question of which of the two numbers described is larger is simply independent of our axioms, whatever they may be. That is, the question of whether one description describes a larger number than another or not may simply be neither provable nor refutable in whatever axiomatic system we have chosen to work in, no matter how strong. In such a case, I would find it to be at least a debatable philosophical question whether there is indeed a fact of the matter about whether one of the numbers is larger than the other or not. (See my explanation here). It is my further considered belief that every largest-number contest, taken seriously, unless severely limited in the scope of the descriptions that entries may employ, will founder on precisely this kind of issue.

For further reading on this topic, there are a number of online resources:

Googol plex bang stack hierarchy | Largest number contest on MathOverflow

Scott Aaronson on Big Numbers (related: succinctly naming big numbers)

Meanwhile, lets return to our actual contest, for which we collected a number of interesting submissions, on which I’ll comment below. Several students had appreciated the connection between the contest and Berry’s paradox.

Your number is quite large, since we can surely describe an enormous number of digits of $\pi$. Researchers have announced, for example, that they have now found 10 trillion digits of $\pi$. If by “describe physically” you meant that we should say the digits aloud, however, then supposing that we speak very quickly and say five digits per second (which is quite a lot if sustained for any length of time), then one person would be able to pronounce a little over 4 million digits this way in 10 days. If all humanity were to speak digits in parallel, then we could rise up to the order of $10^{16}$ digits in ten days. This is a big number indeed, although not nearly as big as a *googol* or many of the other numbers we have discussed.

Surely $\infty$ is a large number, but it is not ordinarily considered to be a positive integer or a natural number. And so I think that your submission does not accord with the rules.

This is charming, and I’m sure that your girlfriend will be very happy to see this. Depending on the units we use to measure your affection, I find it very likely that this number is extremely large indeed, and perhaps you win the contest. (But in order to stay within the rules, we should pick the units so that your affection comes out exactly as a positive integer; in particular, if your affection is actually infinite, as you seem to indicate, then I am afraid that you would be disqualified.) One concern that I have, however, about your delightful entry—please correct me if I am wrong—is this: did you write at first about your affection for your “girlfriends” (plural), and then cross out the extra “s” in order to refer to only one girlfriend? And when you refer to your “girlfriend(s), which values at least $\omega$,” do you mean to say that your affection for each girlfriend separately has value $\omega$, or are you instead saying that you have affection for $\omega$ many girlfriends? I’m not exactly sure. But either interpretation would surely lead to a large number, and it does seem quite reasonable that one might indeed have more affection, if one had many girlfriends rather than only one! I suppose we could accept this entry as a separate submission about your affection, one value for each girlfriend separately, but perhaps it may be best to combine all your affection together, and consider this as one entry. In any case, I am sure that you have plenty of affection.

You’ve submitted Graham’s number, an extremely large number that is also popularly known as the largest definite number ever to be used in a published mathematical proof. This number is certainly in the upper vicinity of the numbers we discussed above, obtained by iterating the Knuth uparrow. It is much larger than anything we can easily write in the *googol plex bang stack* hierarchy. This is essentially in line with iterating the $\Uparrow$ operator $64$ times, using base $3$. This number is extremely large, and I find it to be the largest number submitted by means of a definite description whose numerical value does not depend on which other numbers have been submitted to the contest. Meanwhile, we discussed some even larger numbers above, such as the triple uparrow $10 \uparrow\hskip-.55em\Uparrow$ *googol.*

Your clever submission shows that you realize that it isn’t necessary to actually mention the largest *possible* number to describe, as we discussed in Berry’s paradox above, but it suffices rather to exceed only the largest *actually occurring* numbers that are described by others in the contest. What are we to do, however, when two people have this same clever idea? If either of them succeeds in describing a number, then it would seem that they both must succeed, but in this case, they must both have described numbers larger than the other. This is the same contradiction at the heart of Berry’s paradox. Another risky feature of this entry is that it presumes that some numbers were in fact described by others. How are we to interpret it if all other submissions are disqualified?

Your description has the very nice feature that it avoids the self-reference needed in the previous description to refer to all the “other” entries (that is, all the entries that are not “this” one). It avoids the self-reference by speaking of “second-largest” entry, which of course will have to be the largest of the other entries, since this entry is to be one larger. Although it avoids self-reference, nevertheless this description is what would be called *impredicative*, because it quantifies over a class of objects (the collected entries) of which it is itself a member. Once one interprets things as you seem to intend, however, this entry appears to amount to essentially the same description as the previous entry. Thus, the problematic and paradoxical situation has now actually occurred, making it problematic to conclude that either of these entries has succeeded in describing a definite number. So how are we to resolve it?

Your description is very clever, trying to have it both ways. Of course, every participant in the contest wants to win, but you also to win with a very low winning number, so that you can maximize the prize payout. With this submission, of course you hope that all other submissions might be disqualified, in which case you will win with the number 1, achieving the million dollar payout! In the case of our actual contest, however, there are other cards that are qualified, such as the Graham’s number submission, and so your submission seems to be in case 2, making it similar in effect to the previous submission. Like the previous two entries, this card is problematic when two or more players submit the same description. And indeed, even with the other descriptions submitted above, we are already placed into the paradoxical situation that if this card describes a number in its case 2, then it will be larger than some numbers that are larger than it.

Your submission is very ambitious, attempting both to win and to win the most money. (I take the description as the lower part of this card, and the upper note is a remark about it.) You seem to realize that there will be ties in the game, with multiple people submitting a winning entry. And indeed, if everyone had submitted this entry, then the winning number would indeed seem to be 1, which is the smallest number that would make them all be one of the members who win the game, so the million dollars would be split amongst that group. Some clumsy person, however, might submit a *googol plex bang*, however, which would bump up the value of this card to that same value, which although still winning, would win much less money. Your line of reasoning about rationality is very similar to the concept of superrationality defended by Douglas Hofstadter in the context of the prisoner’s dilemma. Perhaps one might also be tempted to say something like:

*the smallest number that makes me win, with the largest possible prize payout to me.*

with the idea that you’d get a bigger prize solely for yourself if you avoided a tie situation by adding one more, so as to get the whole prize for yourself. But if this reasoning is correct, then one would seem to land again in the clutches of the Berry paradox.

To my readers: please make your submissions or comments below!

The latest installment of math for six-year-olds…

This morning once again I went into my daughter’s first-grade classroom, full of inquisitive six-and-seven-year-old girls, and made a mathematical presentation on the game of Nim.** **

** Win at Nim!**

The game of Nim, I explained, begins with one player setting up a number of stacks of blocks,while the opponent chooses whether to go first or second. Taking turns, each player removes one or more blocks from a stack of their choosing. (It is fine to take a whole stack on your turn.) The player who takes the last block wins.

We demonstrated the game by playing a number of exhibition rounds, and then the girls divided into pairs to play each other and also me. They were surprised that I was able to win against them every single time. In explanation, I told them that this was because in the game of Nim, there is a *super-secret mathematical strategy!* Did they want to learn? Yes! I took as my goal that they would all learn the Nim strategy, so that they could go home and confound their parents by beating them again and again at the game.

Since this was a first-grade class, we concentrated at first on games with stacks of heights 1, 2 and 3 only, a special case of the game which can still challenge adults, but for which six-year-olds can easily learn the winning strategy.

After gaining some familiarity with the game by playing several rounds amongst each other, we gathered again for the secret strategy session. We began by thinking about the case of a game of Nim with only two stacks. They had noticed that sometimes when I played them, I had made copying moves; and indeed I had purposely said, “*I copy you,*” each time this had occurred. The copying idea is surely appealing when there are only two stacks. After some discussion, the girls realized that with just two stacks, if one played so as to equalize them, then one would always be able to copy the opponent’s move. In particular, this copying strategy would ensure that one had a move to make whenever the opponent did, and so one would win the game.

In short order, the girls also realized that if one had any number of pairs of such balanced stacks—so that every stack had a partner—then the whole position was also winning (for one to give to the other player), since one could copy a move on any stack by making the corresponding move on the partner stack. Thus, we deduced that if we could match up stacks of equal height in pairs, then we had a winning strategy, the strategy to copy any move on a partner stack.

In particular, this balancing idea provides a complete winning strategy in the case of Nim games for which all stacks have height one or two. One should play so as to give a *balanced* position to one’s opponent, namely, a position with an even number of stacks of height one and an even number of stacks of height two. Any unbalanced position can always be balanced in this way, and any move on a balanced position will unbalance it.

To handle positions with stacks of height three, the super-secret trick is that one can balance a stack of height three either with another stack of height three, of course, *but also* with two stacks: one of height one and one of height two. Thus, one should regard a stack of height three as consisting of two sub-stacks, one of height one and one of height two, for the purposes of balancing. Thus, the Nim position 1+2+3 counts as balanced, since the 3 counts as 2+1, which balances the other stacks. The 1+2+3 position has two stacks of height two and two of height one, when one regards the stack of height three as having a substack of height two and a substack of height one.

In this way, one arrives at a complete winning strategy for Nim positions involving stacks of height at most three, and furthermore, this is a strategy that can be mastered by first-graders. The strategy is to strive to balance the position. If you always give your opponent a balanced position, then you will win! Faced with an unbalanced position, you can always find a balancing move, and any move on an balanced position will unbalance it. If the game is just starting, and you are deciding whether to go first or second, you should determine whether it is balanced yet or not. If it unbalanced, then you should go first and make the balancing move; if it is already balanced, then you should go second and adopt the copying strategy, in which you re-balance the position with each move.

More advanced players will want to consider Nim positions with taller stacks than three, and we talked about this a little in the classroom. Some of the girls realized that the copying strategy and the idea of balanced positions still worked with taller stacks. One can balanced stacks of height four against other stacks of height four, and so one, but the trick for these taller stacks is that one may balance 5 with 4+1; balance 6 with 4+2; and 7 with 4+2+1. Mathematicians will recognize here the powers of two.

To teach the strategy to children, it is a great opportunity to talk about the powers of two. Any child knows how to count 1, 2, 3, 4 and so on, and most can count by twos 2, 4, 6, 8, 10, …; by fives 5, 10, 15, 20, …; by tens, by threes; by sevens; and so on. , The powers of two are the numbers 1, 2, 4, 8, 16, 32, 64, 128, and so on, doubling each time. Climbing this exponential growth, children are often amazed at how quickly one reaches very large numbers:

One plus one is two;

two plus two is four;

four plus four is eight;

eight plus eight is sixteen;

sixteen plus sixteen is thirty-two;

thirty-two plus thirty-two is sixty-four;

sixty-four plus sixty-four is one hundred twenty-eight.

For Nim, we don’t in practice need such big powers of two, since one doesn’t usually encounter stacks of height eight or larger, and usually just 1s, 2s and 4s suffice. The relevant fact for us here is that every natural number is uniquely expressible as a sum of distinct powers of two, which of course is just another way of talking about binary representation of a number in base two. We regard a Nim stack as consisting of its power-of-two substacks. Thus, a stack of height 3 counts as 2+1; a stack of height 5 counts as 4+1; a stack of height 6 counts as 4+2; and a stack of height 7 counts as 4+2+1.

Ultimately, the winning general strategy for Nim is always to play so as to balance the position, where one regards every stack as being composed of its power-of-two sub-stacks, and a position counts as balanced when these stacks and sub-stacks can be matched up in pairs. This is a winning strategy, since every unbalanced position can be balanced, and any move on a balanced position will unbalance it. To balance an unbalanced stack, play on any stack containing the largest size unbalanced power of two substack, and reduce it so as to balance the parity of all the stacks. If one thinks about it, at bottom what we are doing is ensuring that if we represent the stack heights in their binary representation, then we should play so as to ensure that the position has a even number of one digits in each place.

Today I went into my daughter’s first-grade classroom, full of six-year-old girls, and gave a presentation about Möbius bands.

We cut strips of paper and at first curled them into simple bands, cylinders, which we proved had two sides by coloring them one color on the outside and another color on the inside. Next, we cut strips and curled them around, but added a twist, to make a true Möbius band.

These, of course, have only one side, a fact that the children proved by coloring it one color all the way around. And we observed that a Möbius band has only one edge.

We explored what happens with two twists, or more twists, and also what happens when you cut a Möbius band down the center, all the way around.

It is very interesting to cut a Möbius band on a line that is one-third of the way in from an edge, all the way around. What happens? Make your prediction before unraveling the pieces–how many pieces will there be? Will they be all the same size? How many twists will they have?

Overall, the whole presentation was a lot of fun. The girls were extremely curious about everything, and experimented with additional twists and additional ways of cutting. It seemed to be just the right amount of mathematical thinking, cutting and coloring for a first-grade class. To be sure, without prompting the girls made various Möbius earrings, headbands and bracelets, which I had to admit were fairly cool. One girl asked, “is this really mathematics?”

It seems I may be back in the first-grade classroom this spring, and I have in mind to teach them all how to beat their parents at Nim.