I think I found a bug in a Henry Dudeney book.
Dudeney was a really famous puzzle creator in Victorian/Edwardian times. For Americans: Sam Loyd was sort of an American knock-off of Dudeney, except that Loyd stole half his puzzles from other people and HD didn’t. Dudeney got so annoyed by this theft that he eventually ended up comparing Loyd to the Devil, which was tough talk in 1910.
Anyway, he wrote a number of puzzle books, and at least some are available on Project Gutenberg, so well done the PG people. If you like puzzles, maths or thinking sorts, then there are a few good collections (and there are nicer to read versions at the Internet Archive too). The Canterbury Puzzles is his most famous work, but I’ve been reading Amusements in Mathematics. In there he presents the following puzzle:
81.—THE NINE COUNTERS.
I have nine counters, each bearing one of the nine digits, 1, 2, 3, 4, 5, 6, 7, 8 and 9. I arranged them on the table in two groups, as shown in the illustration, so as to form two multiplication sums, and found that both sums gave the same product. You will find that 158 multiplied by 23 is 3,634, and that 79 multiplied by 46 is also 3,634. Now, the puzzle I propose is to rearrange the counters so as to get as large a product as possible. What is the best way of placing them? Remember both groups must multiply to the same amount, and there must be three counters multiplied by two in one case, and two multiplied by two counters in the other, just as at present.
In this case a certain amount of mere “trial” is unavoidable. But there are two kinds of “trials”—those that are purely haphazard, and those that are methodical. The true puzzle lover is never satisfied with mere haphazard trials. The reader will find that by just reversing the figures in 23 and 46 (making the multipliers 32 and 64) both products will be 5,056. This is an improvement, but it is not the correct answer. We can get as large a product as 5,568 if we multiply 174 by 32 and 96 by 58, but this solution is not to be found without the exercise of some judgment and patience.
But, you know what? I don’t think he’s right. Now, I appreciate that he probably had to spend hours or days trying out possibilities with a piece of paper and a fountain pen, and I just wrote the following 15 lines of Python in five minutes, but hey, he didn’t have to bear with his government trying to ban encryption, so let’s call it even.
from itertools import permutations nums = [1,2,3,4,5,6,7,8,9] values =  for p in permutations(nums, 9): one = p*100 + p*10 + p two = p*10 + p three = p*10 + p four = p*10 + p if four > three: continue # or we'll see fg*hi and hi*fg as different if one*two == three*four: expression = "%s*%s = %s*%s = %s" % ( one, two, three, four, one*two) values.append((expression, one*two)) values.sort(key=lambda x:x) print("Solution for 1-9") print("\n".join([x for x in values]))
The key point here is this: the little programme above indeed recognises his proposed solutions (
158*32 = 79*64 = 5056 and
174*32 = 96*58 = 5568) but it also finds two larger ones:
584*12 = 96*73 = 7008 and
532*14 = 98*76 = 7448. Did I miss something about the puzzle? Or am I actually in the rare position of finding an error in a Dudeney book? And all it took was seventy years of computer technology advancement to put me in that position. Maths, eh? Tch.
It’s an interesting book. There are lots of money puzzles, in which I have to carefully remember that ha’pennies and farthings are a thing (a farthing is a quarter of a penny), there are 12 pennies in a shilling, and twenty shillings in a pound. There’s some rather racist portrayals of comic-opera Chinese characters in a few of the puzzles. And my heart sank when I read a puzzle about husbands and wives crossing a river in a boat, where no man would permit his wife to be in the boat with another man without him, because I assumed that the solution would also say something like “and of course the women cannot be expected to row the boat”, and I was then pleasantly surprised to discover that this was not the case and indeed they were described as probably being capable oarswomen and it was likely their boat to begin with! Writings from another time. But still as good as any puzzle book today, if not better.