## The number that must not be named

*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…