Lets take a number 12 as an example .
It can be partitoned into three integer summands in 12 ways:

12=1+1+10

12=1+2+9

12=1+3+8

12=1+4+7

12=1+5+6

12=2+2+8

12=2+3+7

12=2+4+6

12=2+5+5

12=3+3+6

12=3+4+5

12=4+4+4

Multiplying the 3 members of each partitions results in 12 **distinct** numbers: 10,18,...60,64.

On the other hand the same treatment applied to number 13 produces a pair of **equal **results: 13=1+6+6=2+2+9 and 1*6*6=2*2*9=36 (a well known problem of children's ages).

Find the smallest number which has 3 distinct partitions into 3 parts, each of them with the **same** product.

Bonus: list all numbers below 1000 boasting this feature.

Good scales have equal arms (arms are the the things that connect the actual scale to the center), but in one grocery stall, the arms of the scale are not equal. Pending replacement, the manager wonders if he can give correct weight this way:

"I'll balance a 1-pound weight on the left with sugar on the right, and then I'll balance the 1-pound weight on the right with some more sugar on the left, and the sugar will add up to exactly 2 pounds."

Will it? What are other (assuming that the above works, it may not) ways of weighing 2 pounds of sugar, if you also have a lead shot with you to help weigh? (Note and hint: The lead shot has an unknown weight. You can make it whatever weight you choose. Remember, the arms aren't equal, and you need 2 pounds of sugar.)