Counting To 52

fluffy brought to my attention a cute, amusing little bit from the Annals of Improbable Research, itself passing on some work by one Inder J Taneja. Taneja worked out a paper, available from arxiv.org, which lists results to the sort of mathematical puzzle that’s open to anyone with some paper and a pencil and some desire to do some recreational stuff.

Specifically, he starts with the digits 1 through 9, in ascending and then descending order. What numbers can be made by operations which combine them in order? By this he means taking the numbers and adding them, multiplying them, taking exponentials, grouping them parenthetically, or making two or more digits into a two- or more-digit number. That is, for example, 2 and 3 could be used for 2 + 3, for 2 x 3, for 2³, for (2 + 3), or for the number 23.

Taneja found sequences of additions like this to provide all the numbers from 44 to 1,000, excepting 52. Some of them are pretty slick; for example, 623 is (12 + 3) x 4 + 5 + (6 + 7 x 8) x 9, which would have taken me forever to figure out. And some look a bit like cheating: Taneja works out 346 as 1²³⁴⁵ + 6 x 7 x 8 + 9, although I admit I probably wouldn’t have worked out a better way to do that one.

He also works them out for descending orders, although there are more gaps here: he hasn’t found any for 47, 51, 52, 53, 58, 61, 62, or 70, although he has some lovely constructs like 964 equalling (9 + 87 + 6 + 5) x (4 + 3 + 2) + 1.

The obvious question would be why not allow subtraction or division for this? Taneja doesn’t say, although I’d guess that the goal was to make the puzzle as challenging as possible — thus the limits on how to combine numbers — without making it impossible to make very many numbers. This probably won’t lead to any important mathematics, but it’s well-placed for playing; note that Taneja claims to have worked out all these without turning to the computer for calculations.

And I do wonder about questions like what number can be made the most ways from this sort of combination, or what the distribution of numbers which can be made is like. For that matter I’m not immediately convinced I know how what the biggest constructible number is, or what’s the smallest number that can be written in descending order.