September 27th, 2003

s60 harriet half smile

Problem Solving class

Is there enough information to solve this problem?

"The clown speaks to the magician: 'I lent some money to five people here and still haven't been paid back. You're one of them; the other four owe me twelve dollars altogether, but I don't remember how much each of them owes me separately.'

'Whole dollars, no cents?'

'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 can't figure out how much each of the other four owe you.'

'Wait; the lion tamer is the one who owes me the least.'

'That'll do it! Now I know the amount of each debt.'

How much did each of the five people owe the clown?"


I don't see how there is enough info to have only one possible answer. I mean, it could be that the debts are 1, 2, 4, 5, and 40, or it could be that the debts are 3, 4, 4, 5, and 240. All I seem to realize that I know is that one debt is smaller than all the others, that four added together equal 12 and that the product of those same four is the fifth. I don't see how that is enough--am I wrong?
angelina

Solution

Thanks to tiwonge for pointing out the piece of info I was missing and for doing it in a way that I understood.

Before the magician knows that one is lower than all the others, he knows what his debt is. So, we want to look at sets that add to 12 where the lowest number is repeated at least once. Those are as follows, with the products in paranthesis after the set:

1,1,1,9 (9)
1,1,2,8 (16)
1,1,3,7 (21)
1,1,4,6 (24)
1,1,5,5 (25)
2,2,2,6 (48)
2,2,3,5 (60)
2,2,4,4 (64)
3,3,3,3 (81)

Those are the possibilities so far. Then the magician is told that the lion tamer's is the lowest. So, we know that the lowest number does not repeat. What are the possiblities without the lowest number repeating?

1,2,3,6 (36)
1,2,4,5 (40)
1,3,3,5 (45)
1,3,4,4 (48)
2,3,3,4 (72)

Now, you know that before and after the magician knew his debt, which means that there should be one set from each where the products are equal. The only ones that have an equal product are 48 (2,2,2,6 and 1,3,4,4)

So the debts are: Lion tamer owes $3, magician owes $48, and the other three owe $3, $4 and $4. Course if I were the magician I would go with the 1,2,3,6 set just cause it means that he owes less. Brett was wrong; never trust magicians, not clowns.