I have written on a number of occasions about the book 'Calculator Fun and Games' by Ben Hamilton, which my parents gave to me as a Christmas present in the mid 1980s. I'd always found maths interesting, but this book (and a calculator) opened up a world of puzzles and fun in maths. I have revisited this book periodically, mining it for blog articles, and here I am again, going through it page by page looking at the underlying maths behind some of the games and puzzles it contains.
One of them is known simply as 'Crafty Calculator Calculations', and is a series of multiplications, with the question, "What do you notice about these numbers?"
Here are the multiplications. I shall resist the temptation to throw them all into a spreadsheet, and in the spirit of the book will be using my calculator (a Casio fx-85MS which is about 20 years old).
Section 1: The answers as written
2 * 8714 = 17428
2 * 8741 = 17482
3 * 4128 = 12384
3 * 4281 = 12843
3 * 7125 = 21375
3 * 7251 = 21753
6 * 7251 = 43506 (?)
8 * 4973 = 39784
8 * 6521 = 52168
9 * 7461 = 67149
14 * 926 = 12964
24 * 651 = 15624
42 * 678 = 28476
51 * 246 = 12546
78 * 624 = 48672
87 * 435 = 37845
75 * 231 = 17325
65 * 281 = 18265
65 * 983 = 63895
72 * 936 = 67392
What do I notice? With the exception of a probably typo, each of the answers is a numerical anagram of the two numbers being multiplied together.
Extension work 1: Fixing the typo
Firstly, 6 * 7251 = 43506, which is not a numerical anagram.
Hypothesis: the 7251 is a typo, having appeared in the previous line. The six is probably correct, since it comes between a series of threes and before a pair of eights.
I've tried multiplying 7251 by 4, 5, 6 and 7, and none of them produce numerical anagrams, so the 7251 is definitely the typo.
Hypothesis: Fixing the typo is going to be difficult, unless I can find some patterns in the larger of the two numbers which are being multiplied. Is there a pattern?
What do these numbers have in common?
8741 - prime
8714 - not prime
7461 - not prime
926 - not prime
So the answer is - not much. It's more like a random distribution, although 1 and 7 appear quite frequently (because any number y multiplied by another number that ends in 1 will have the digit y at the end).
I can find (with relative ease) examples where I can get an anagram of four of the five digits:
5 * 6321 = 31605 (containing a zero instead of a two)
6 * 5321 = 31926 (containing a nine instead of a five)
But I haven't yet found one that will replace the error in the original question.
So the question is: Is the initial set a complete set with no further triplets of numbers?
It's unlikely that this is a complete set - they're such an arbitrary bunch that it's highly possible there are many more. But is there a way of making this a systematic search instead of just random guessing?
- the units of the larger number are either 1, 3, 4, 5, 6 or 8
- the smaller number and larger number have to multiply together to form a five-digit number. 3 * 2894 will only produce another four-digit number, for example.
The answer (through only trial and improvement) is that this is NOT a complete set of triplets. For example:
5 * 2519 = 12595 is an example not included in the original list.
Four digits: 27 * 81 = 2187 (that's what you get for asking AI for help, and not specifying that you want a five-digit number).
Conclusions:
AI says this is an interesting exercise in combinatorics, but didn't actually add anything to the search (it would have taken longer to explain the question to it than it would to have found more myself).
There is a specific quantity of numbers that satisfy the requirement a *b = c where c is a numerical anagram of a and b, containing each of the digits of a and b once each, and where c is a five-digit number.
Can I fix the typo? No, and I have tried!
