## Wednesday, March 14, 2007

### Pocket Change

A friend of yours (me!) announces he has a number of coins in my pocket that add up to precisely one dollar. When I tell you how many coins I have, you ask if any one of them is a half-dollar. I tell you no. You quickly realize that you can't tell for sure what coins I have, because there are six different combinations that produce precisely one dollar.

How many coins do I have in my pocket?

1. It looks like, when you divide the number of nickels by 3, the remainder is 1. That's interesting. Five nickels and ten dimes makes a dollar, and there are five more ways with the same number of coins (as long as half dollars are excluded)

2. Oops! I mixed up the words "nickels" and "dimes" above. All else is good.

3. That's right Jonathan. You need 15 coins.
Since there are no half dollars, you can use quarters, dimes, nickels and pennies. The six ways you can do this are:
__Q D N P
1)3 1 1 10
2)2 1 7 5
3)1 1 13 0
4)1 5 4 5
5)0 9 1 5
6)0 5 10 0

And this is a unique answer, although I'll leave the proof to you ;-)

4. maybe im missing somthing? but four quarters make a dollar, ten dimes make a dollar, twenty nickels make a dollar, and one hundred pennies make a dollar. thats all common knowledge, but none of the above use fifteen coins, and adding to the above combinations, that makes ten different combinations?

5. But how do you like nickels/3 has a remainder of 1 detail? 1, 4, 7, 10 or 13.

One idea here is that we are trading coins such that we have equal numbers and equal values. How can we do that?

4D = 1Q3N
1Q5P = 6N
------------
1Q4D5P = 1Q9N
-1Q -1Q
------------
4D5P = 9N

So the first two lines can be combined to create the last. We just need enough coins to apply this 5 times.

6. jonathan, there are some days when I think I shouldn't be the one running this. Seeing your answer makes me think that way.

Nick, the idea for this question was to figure out how many coins you needed in your pocket so that there are six ways (and only six ways) of making one dollar. Four quarters make up a dollar, but there is no other combination of four that makes up a dollar.

7. Mike,

you come up with these every day? I know I disappear for weeks on end (rhythms of life and teaching). I am happy when I have a nice solution with a reasonable explanation.

But you do this every day. Credit where credit is due. No way could I do this.

Btw, have you looked at my (far less frequent) puzzles?

8. Jonathan, I have looked at them from time to time. I like your recent puzzle on black and white marbles and thought about creating a variant here. But, what with time limits, I just haven't gotten a chance.

I really appreciate the encouragement, thanks.

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