If a bunch of positive integers adds up to 20, what is the greatest possible product of those numbers?
There are two strategies to try here: use the largest numbers and multiply them together, or grab a bundle of smaller numbers and multiply those together instead.
1. Large numbers
10+10 = 20
10 * 10 = 100
9+9+2 = 20
9 * 9 * 2 = 162
A reasonable start. Let's try some more at the other end of the scale - the smaller numbers.
2. Small numbers
5 * 5 * 5 * 5 = 625
4 * 4 * 4 * 4 * 4 = 1024
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024
Moving towards the smaller numbers has definitely yielded larger products, but we've identifed a maximum between 4^5 and 2^10. There may be a winner between the two.
Indeed, 2 * 3^6 gives...
3 * 3 * 3 * 3 * 3 *3 * 2 = 1458
...which is the highest total product of digits which sum to 20.
There are two strategies to try here: use the largest numbers and multiply them together, or grab a bundle of smaller numbers and multiply those together instead.
1. Large numbers
10+10 = 20
10 * 10 = 100
9+9+2 = 20
9 * 9 * 2 = 162
A reasonable start. Let's try some more at the other end of the scale - the smaller numbers.
2. Small numbers
5 * 5 * 5 * 5 = 625
4 * 4 * 4 * 4 * 4 = 1024
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 1024
Moving towards the smaller numbers has definitely yielded larger products, but we've identifed a maximum between 4^5 and 2^10. There may be a winner between the two.
Indeed, 2 * 3^6 gives...
3 * 3 * 3 * 3 * 3 *3 * 2 = 1458
...which is the highest total product of digits which sum to 20.
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