My continued mining of the paperback Calculator Fun and Games by Ben Hamilton continues. My previous post looked at multiplication calculations where the product of a multiplication contained the same digits as the two numbers being multiplied; for example: 2 * 8741 = 17482 leading to numerical anagrams. In this post, I move forwards a few pages through the book to a similar set of questions, along with the question, "What do you notice?"
Calculators at the ready: here are some more multiplications:
Here goes:
8 * 473 = 3784
9 * 351 = 3159
15 * 93 = 1395
21 * 87 = 1827
27 * 81 = 2187
35 * 41 = 1435
In this case, they're all four-digit numerical anagrams. I mentioned in previous post that my conversations with AI led me to find a four-digit numerical anagram of this kind - mine was the 27 * 81 = 2187. So, are there any more?
Yes: Copilot has found a few (possibly through brute force?)
21*60 = 1260
30*51 = 1530
80*86 = 6880
These are the two-digit * two digit pairs.
There are some single-digit * three-digit pairs that can be added to the list:
3* 501 = 1503
3* 510 = 1530
5* 251 = 1255
6* 201 = 1206
6* 210 = 1260
8 * 860 = 6880
Copilot has indicated that these are ALL the pairs; and it can also suggest a Python or BASIC loop that tries all the combinations of digits in turn (brute force). It doesn't, however, provide much output on identifying patterns in the numbers. It helpfully points out that a is a single-digit number, and the other number, b, is a three-digit number. Their product, a *b is a four-digit number.
I'd call out 6880, which features twice (8*860 and 80*86). However, that's only because it's possible to move the zero between either a or b. The same applies to 3*510 (30*51) and 6*210 (60*21).
Otherwise, there's no obvious pattern - these numbers only produce numerical anagrams because of a certain coincidence that occurs in base 10.
15 * 93 = 1395
21 * 87 = 1827
27 * 81 = 2187
35 * 41 = 1435
In this case, they're all four-digit numerical anagrams. I mentioned in previous post that my conversations with AI led me to find a four-digit numerical anagram of this kind - mine was the 27 * 81 = 2187. So, are there any more?
Yes: Copilot has found a few (possibly through brute force?)
21*60 = 1260
30*51 = 1530
80*86 = 6880
These are the two-digit * two digit pairs.
There are some single-digit * three-digit pairs that can be added to the list:
3* 501 = 1503
3* 510 = 1530
5* 251 = 1255
6* 201 = 1206
6* 210 = 1260
8 * 860 = 6880
Copilot has indicated that these are ALL the pairs; and it can also suggest a Python or BASIC loop that tries all the combinations of digits in turn (brute force). It doesn't, however, provide much output on identifying patterns in the numbers. It helpfully points out that a is a single-digit number, and the other number, b, is a three-digit number. Their product, a *b is a four-digit number.
I'd call out 6880, which features twice (8*860 and 80*86). However, that's only because it's possible to move the zero between either a or b. The same applies to 3*510 (30*51) and 6*210 (60*21).
Otherwise, there's no obvious pattern - these numbers only produce numerical anagrams because of a certain coincidence that occurs in base 10.
It's still fun, though!
Previous Calculator Fun and Games articles:
Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams, five digits)
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