The Number Mysteries: A Mathematical Odyssey Through Everyday Life, by Marcus du Sautoy

Second paragraph of third chapter:

From ancient civilizations all around the world we have a fascinating assortment of games. Stones thrown in the sand, sticks tossed in the air, tokens placed in hollows carved into wooden blocks, hands used to compete, pictures drawn on cards. From ancient mancala to Monopoly, from the Japanese game of go to the poker tables of Vegas, games are invariably won by whoever is best at taking a mathematical, analytical approach. In this chapter I will show you how maths is the secret to the winning streak.

I read another of du Sautoy’s books twenty years ago in my early book-blogging days. This one is a straightforward romp through various bits of mathematical theory – prime numbers, topology, probability, cryptography and dynamics. I didn’t learn a lot from it, but it is breezily done and will probably appeal to smart older kids who are presumably the target audience. You can get it here.

This was my top unread book acquired in 2016. Next on that pile is Deuces Down, edited by John Jos. Miller.

Square numbers ending in repeated digits

A mathematical curiosity that I have been thinking about.

You have probably noticed that 12², in base 10, is 144. (Actually the square of the number 12 in any base from 5 upwards is always 144 in that base, for fairly obvious reasons.) You might not have noticed that 38², in base 10, is 1,444. I did notice this and wondered if it’s possible to find squares with ever increasing numbers of 4’s at the end.

The short answer is no. The long answer is that, apart from the obvious case of square numbers ending in 0, if you are sticking to base 10 the most repeated digits you can have at the end is three 4’s, as in 38². All square numbers in base 10 ending in 1 have an even number in the tens column. (01, 81, 121, 361…) All square numbers in base 10 ending in 9 have an even number in the tens column. (09, 49, 169, 289…) All square numbers in base 10 ending in 5 have a 2 in the tens column. (25, 225, 625…)

All square numbers in base 10 ending in 4, however do have an even number in the tens column (04, 64, 144, 324…) so one in five of them will have two 4’s at the end. Square numbers in base 10 ending in …44 can have either an odd or an even number in the hundreds column, and it’s not much work to show that any number in base 10 of the form 500n ± 38 (where n is an integer) will have a square that ends in …444. However that is the end of the story, as the number in the thousands column will always be odd: 38² = 1,444; 462² = 213,444; 538² = 289,444; 962² = 925,444 and so on.

That’s in base 10. Other bases are a different matter. (Though again, in every case, you can always get square numbers which end in increasing 0’s.) In base 2, all odd squares end in ..001, so there is no chance of repeated digits at the end other than 0. In base 4, all odd squares end either in …01 or …21.

For odd numbers, however, there seems to be no limit. In base 3, for instance, the number 121 (base 10) the square of 11 (base 10) is 11,111 in base 3 (81+27+9+3+1). In base 5 and above, square numbers that end in 1 can be multiplied by 4 to get more square numbers that end in 4.

And larger even numbers may offer more flexibility too. In base 16 (hexadecimal), the square of 497 (1175 in base 10) is 15,111 (1,380,625 in base 10); the square of DC5 (3,525 in base 10) is BD9,999 (12,425,625 in base 10).

Anyway, that’s as far as I have got for now.