An Accountant Seeks the Help of a Mathematician

The accountant complaints to the mathematician:

“I lent money to five other faculty members and still haven’t been paid back. You are one of them; the other four owe me 12 dollars altogether, but I don’t remember how much each person owes me separately.”

“Are the debts in whole dollars?”

“Yes. I do remember that the four other debts multiplied together equals your debt. Do you remember how much you owe me?”

“Yes, but I still haven’t figured out how much each of the other four owe you.”

“Wait! The statistician is the one who owes the least.”

“That does it. Now I know the amount of each debt.”

What are the debts and how did the mathematician determine them?

The mathematician owes

48 dollars

and the other faculty members owe

4, 4, 3 and 1 dollar.

Explanation:

We know the total debt of the other four faculty members is 12, so we're looking for partitions of 12 in four parts. Let's list them and compute the product (which would be the mathematician's debt):

Table:

9 + 1 + 1 + 1 = 12 --> 9
8 + 2 + 1 + 1 = 12 --> 16
7 + 3 + 1 + 1 = 12 --> 21
7 + 2 + 2 + 1 = 12 --> 28
6 + 4 + 1 + 1 = 12 --> 24
6 + 3 + 2 + 1 = 12 --> 36
6 + 2 + 2 + 2 = 12 --> 48
5 + 5 + 1 + 1 = 12 --> 25
5 + 4 + 2 + 1 = 12 --> 40
5 + 3 + 3 + 1 = 12 --> 45
5 + 3 + 2 + 2 = 12 --> 60
4 + 4 + 3 + 1 = 12 --> 48
4 + 4 + 2 + 2 = 12 --> 64
4 + 3 + 3 + 2 = 12 --> 72
3 + 3 + 3 + 3 = 12 --> 81
Now, the mathematician knows his own debt, so if he's unsure of the distribution of the others, there must be multiple partitions of 12 in four parts with a product equal to his debt. That's only the case for 48, which can be split as $$6 \times 2 \times 2 \times 2$$ and $$4 \times 4 \times 3 \times 1$$. Since there is a single least number among those (corresponding to the statistician's debt), it must be the second option.

The mathematician owes

\$48 Because nothing is said that the individual debts are unique, only that the lowest one must be. So any of the following distributions of debts by the four remaining are possible, and thus the following possibilities for what the mathematician owes: 1,2,2,7 → \$28
1,2,3,6 → \$36 1,2,4,5 → \$40
1,3,3,5 → \$45 1,3,4,4 → \$48

Now the mathematician says at first that

the other totals are not known to them until the accountant mentions that the statistician owes the least, thus requiring there to be a distinct "least". Prior to that discovery, ...

The mathematician can't owe \$45 because there is only one way to sum to 12 and multiply to 45. The mathematician can't owe \$36 because there is only one way to sum to 12 and multiply to 36.
The mathematician can't owe \$28 because there is only one way to sum to 12 and multiply to 28. The mathematician could owe \$48 because 2,2,2,6 is another valid solution that sums to 12 and multiplies to 48, but is ruled out when the discovery is made.

• In the second part, I think you are missing the possibility 2,3,3,4 – Cain Mar 14 at 21:56