The Only Number That Describes Itself

Here’s a classic brainteaser that I don’t like. What’s the next number in this sequence: 1, 11, 21, 1211, 111221, …? The answer is 312211, because each number describes the digits in the number that precedes it. We open with 1, an arbitrary choice, but the next number describes 1 as “a single one,” i.e. “one one,” i.e. 11. The next entry describes 11 as “two ones,” or 21. This, in turn, is “one two followed by one one,” or 1211, and so on.

Legendary mathematician John Conway studied this so-called “look-and-say” sequence and actually proved some interesting results about it. It clearly goes on forever, and the numbers grow to infinity, but surprisingly no digits other than 1, 2, and 3 ever appear. If you keep describing bigger and bigger numbers in this way, you’ll never generate a string of four ones (or twos or threes) in a row. Conway also studied the sequences that spring from different starting numbers other than 1. He proved that no matter what whole number you open with, the resulting sequence will diverge to infinity… except for one. Determining which one is your bonus puzzle this week.

I like the idea of numbers describing other numbers, but I’d prefer it not cast as a puzzle to solve. My gripe with sequence puzzles is that they’re open to multiple possible solutions. You could surely cook up some strange mathematical operation that produces the same first five numbers as the look-and-say sequence but then deviates from there. Your main puzzle this week concerns a number that describes itself. And rest assured it has only one solution.

Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!

Puzzle #39: A Self-Referential Number

Only one 10-digit number has the following property. Its left-most digit is the number of 0s in the number, the next digit is the number of 1s in the number, the next is the number of 2s, and so on until the right-most digit, which is the number of 9s in the number. Find the number. Numbers can’t begin with a zero.

An example of a four-digit number with this property is 2020. The first digit indicates that the number contains two 0s, the next indicates zero 1s, the next indicates two 2s, and the final indicates zero 3s.

Bonus: you can seed the look-and-say sequence with any whole number. For example, if you started with 39, then the next entry would be 1319 (one three, one nine). Conway proved that all seeds yield a sequence whose entries grow to infinity, with only one exception. Find the exception.

I’ll be back next Monday with the solutions and a new puzzle. Do you know a cool puzzle that you think should be featured here? Message me on X @JackPMurtagh or email me at gizmodopuzzle@gmail.com

Solution to Puzzle #38: Tax Evasion

Shout-out to 8×10 for a swift answer to last week’s tax evasion puzzle. I hope the IRS doesn’t monitor these…

You can win a maximum of $50 in The Taxman Game. See the turns below:

1. You take $11 and the Tax Collector takes $1 (1 is the only available factor of 11)
2. You take $10 and the Tax Collector takes $2 and $5
3. You take $9 and the Tax Collector takes $3
4. You take $8 and the Tax Collector takes $4 ($2 was already taken on move 2)
5. You take $12 and the Tax Collector takes $6
6. You’re out of legal moves so the Tax Collector takes the final check of $7

Your winnings total $8 + $9 + $10 + $11 + $12 = $50.

To save you time trying to hoard even more of Uncle Sam’s due, here’s a little argument that proves the above strategy is optimal. You can only ever take at most one prime numbered paycheck throughout the whole game. Because once you do, the Tax Collector takes the $1 paycheck and all other primes become off limits (the Tax Collector wouldn’t get paid). To avoid the $1 paycheck being wasted on another turn, you must start with a prime number and you might as well make it as big as possible, hence opening with $11.

Now the very end of the game will involve the Tax Collector taking the $7 paycheck no matter what happens, because you can never take it for yourself and no multiples of 7 are available to make the Tax Collector take it earlier. So effectively three paychecks are out of play ($11, $1, and $7), leaving nine remaining. You cannot get more than four of these nine because the Tax Collector must get paid on every turn. The strategy we gave gets you $12, $10, $9, and $8, the four largest remaining paychecks. So our approach cannot be improved.