Listen With Others

An idea entered my head of a puzzle that involved a sequence where each number was formed from the digits of the previous number in some way. I started by summing the squares of the digits of 2- and 3-digit numbers but that got nowhere interesting. Summing the cubes was better, but the longest sequence was 14 (from 177) which was not enough numbers to fill a decent-sized grid. This sequence came to a natural end at 153, because the sum of the cubes of those digits is 153 again.

Ah ha! That is a narcissistic number! (A narcissistic number is one where an n-digit number is equal to the sum of the nth powers of its digits). I first came across narcissistic numbers many years ago when I read that 4679307774 was a ten-digit number that is equal to the sum of the tenth powers of its digits.

I then created sequences based on 4th and 5th powers to see how many terms the longest sequences had before reaching a narcissistic number.

For the 4th powers any sequences of a reasonable length ended with 8208, and for the 5th powers it was 54748 (6th powers would produce numbers which were too large, and so I decided to base the puzzle on just these three sequences). I knew from experience that I needed to have a lot of 3-digit numbers and no more than a couple of 4-digit ones to have any chance of success in populating a symmetrical grid with plenty of checked digits in it.

Now it was time to consider the grid. I started with an 8×8 one, but in order to have a reasonable chance to fill it I had to split the 4- and 5-digit numbers into 2- or 3-digit ones. My constraints were tight: I wanted to fit 16 two-digit numbers, 20 three-digit ones and 2 four-digit ones, with at least half of the digits of each number checked. To avoid an inelegant grid, I didn’t want more than two bars joined in a straight line or more than 11 connected. It proved impossible to create any grids at all even with weaker constraints and so I decided to use a 7×8 grid instead. This immediately looked a lot more promising.

My archaic program (I wrote it over 25 years ago) generated and attempted to fill 2,529,332 grids which fitted my criteria and it was possible for just one of them (the 425,397th) to be populated with numbers from my three sequences. This was very satisfying, especially since no more than eight bars were connected.

What really helped the chance of filling a grid was that the first number of each sequence could be any permutation of its digits and that each 5-digit number could each be split in two ways. As a last resort, I could have added a zero to the first number, but it turned out that I didn’t need to.

Having produced an acceptable grid it was time for the clues.

I resorted to the overworked theme of letter values because it was necessary to provide a hint about the sequences.

Using upper- and lower-case letters I could get quite a good phrase, and I came up with CALCULATE POWERS OF DIGITS, but I didn’t want solvers to anagram the letters to arrive at this and so I added a few extra letters to get CALCULATE POWERS XYZ OF DIGITS which was intended to indicate that more than one power was involved. I didn’t think this was clear enough, and in the end I decided upon TRY SUMMING A POWER OF EACH, with the preamble saying something like ‘…a suggestion will appear regarding the digits of each number…’, hoping that not too many solvers would try anagramming the letters.

I decided that it would be fun (and a challenge) if most of the clues spelled a word. Starting with a glossary of mathematical terms and then adding to it, I ended up with a list of 278 possible words (some of which were the same word but with the case of the letters different) and wrote a program that added mathematical signs to them in all possible ways and then evaluated the resulting expression. I ended up with a long list of possible clues.

I then noticed that I could get NARCISSISTIC NUMS out of the letters – this was completely unintentional! Moreover, I could split this into NARC IS SIS TIC NUMS, and I decided to use these words in the last five down clues (I seem to remember that this took some time). In order to hide this, I added some other expressions to them and these ended up being the starting point for the solving process.

I then manually chose the rest of the clues from the long list so that there was a solving path which used minimal trial and error.

I realised that I had to give the start of one of the sequences to help solvers establish the rule.

Like most of my puzzles, it would not exist without the use of a computer, and having been programming for over 50 years it is an extremely useful tool!

Readers might like to verify, as a bedtime exercise (using ruler and compass only), that the following 39-digit number is narcissistic:

115 132 219 018 763 992 565 095 597 973 971 522 401