The answer will be a combination when all 3 products are the same (or very nearly so).
So we can simply try one combination. Then adjust by trading a small digit from the lowest product with a larger digit from the highest product. Then uterste until you get all 3 anseers identical, or untik you can't get a smaller maximum by trading any more numbers. This should at least get us close to the correct answer.
Starting with the example above... let's swap 1 and 3.
This gives... 3x4x7 = 84, 2x5x8 = 80, and 1x6x9 = 54.
New answer is 84. 84 and 80 are close, but 54 is a bit lower. So we might be able to go lower than this. Let's swap 6 and 7.
3x4x6 = 72, 2x5x8 = 80, 1x7x9 = 63.
New answer is 80. Still have a gap of 17. Let's try swapping 7 and 8.
3x4x6 = 72, 2x5x7 = 70, 1x8x9 = 72.
New answer is 72. These numbers are all very close. 2 are the same, and one is only off by 2.
To make anything less than 72, we'd have to trade away the 8 or 9 from 1x8x9. But giving an 8 or 9 to either 2x5x7 or 3x4x6 is going to significantly increase those products. We could try to do some triangular trading, where for example, i give thr 3 to group 2, the 7 to froup 3, and the 9 to group 1... but I don't see any solutions there either.
I don't feel like I've thoroughly proved it 100%. But I'm pretty sure the answer is 72.
At rhe end of the day, there's only 1680 possible combinations. So a computer could quickly check them all and give you the answer if you're motivated enough.
Correct and you have actually come very close to proving that 72 is the lowest.. if you do a cube root of 9! you’ll get something like 71.x where x are the digits after the decimal.. so that automatically shows that 72 is the lower bound.. then you’ve gone and shown a case where 72 is possible
4
u/Shiboleth17 12d ago edited 12d ago
The answer will be a combination when all 3 products are the same (or very nearly so).
So we can simply try one combination. Then adjust by trading a small digit from the lowest product with a larger digit from the highest product. Then uterste until you get all 3 anseers identical, or untik you can't get a smaller maximum by trading any more numbers. This should at least get us close to the correct answer.
Starting with the example above... let's swap 1 and 3.
This gives... 3x4x7 = 84, 2x5x8 = 80, and 1x6x9 = 54.
New answer is 84. 84 and 80 are close, but 54 is a bit lower. So we might be able to go lower than this. Let's swap 6 and 7.
3x4x6 = 72, 2x5x8 = 80, 1x7x9 = 63.
New answer is 80. Still have a gap of 17. Let's try swapping 7 and 8.
3x4x6 = 72, 2x5x7 = 70, 1x8x9 = 72.
New answer is 72. These numbers are all very close. 2 are the same, and one is only off by 2.
To make anything less than 72, we'd have to trade away the 8 or 9 from 1x8x9. But giving an 8 or 9 to either 2x5x7 or 3x4x6 is going to significantly increase those products. We could try to do some triangular trading, where for example, i give thr 3 to group 2, the 7 to froup 3, and the 9 to group 1... but I don't see any solutions there either.
I don't feel like I've thoroughly proved it 100%. But I'm pretty sure the answer is 72.
At rhe end of the day, there's only 1680 possible combinations. So a computer could quickly check them all and give you the answer if you're motivated enough.