## Friday, May 25, 2012

### Swapping the Roles

Here is a trick that works great in coaching - whether for a math test, spelling test or almost anything else.  Ask your kid to be the teacher and give you the exercises that she/he think are going to be on the test. The hardest ones. While you will be playing a role of a student that solves the test but makes some mistakes. Let your kid check your test and find these mistakes.  Also, let her/him grade you.  Spice it up by playing misbehaving kids, excelling kids, failing kids. This is usually much more fun for them than the regular way of test prep.

I tried this with my daughter this week in preparation for her multiplication test. 20 multiplication exercises. I could solve only half... She liked it so much that next day she prepared a test sheet for me in advance at school and an answer sheet for her to grade me.

Now, let's try doing something similar with our puzzle today. Here is the story. Find what's wrong with it.

You want to send your kids to an 8-week long sleep away summer camp. But you don't have the \$1,997 that it costs. How about asking for help from the grandparents? Each couple offers to give \$1,000 toward the camp. You get the \$2,000, pay for the camp and receive a \$3 change. As a gesture you want to give each of the set of grandparents a \$1 of change, and you keep a \$1 for yourself as \$3 is not divisible by 2.  So, now you owe each couple of grandparents \$999, together you owe them \$1,998. And you have \$1 in your hands.  As you started with \$2,000, this means that \$1 + \$1,998 = \$2,000.  Right?

Disclosure: this puzzle is reinterpretation of a famous missing dollar puzzle that we discussed long time ago.

Dollar image from Flickr, distributed under CCL. Your answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.

Dennis (of Dennis and Katrina) said...

Ah, a famous one!

Originally: \$1000 + \$1000 = \$2000

After payment for camp:
\$1000 + \$1000 - \$1997 = \$3

After refunds:
\$999 + \$999 = \$1997 + \$1 (your dollar).

The problem comes from a logical error in your last line, not math. The \$999 from each set of grandparents and the \$1 you kept should not be compared with the \$2000; by refunding \$2 and keeping \$1, you've changed the problem. The \$999 times 2 and the \$1 should mathematically compare to \$1997, what you paid for the camp:

\$999 + \$999 - \$1997 = \$1.

There is no missing dollar!

Dennis

anne-marie said...

Wrong. The person must give back \$999 to each couple which is 1998 because she aready gave \$1 to each couple so 1998+ 2= 2000

Ilya said...

Yes, this looks familiar, but my rusty old brain does not have the ready explanation, so I have to construct it again :-). The problem is with trying to add quantities that have different meanings. Let’s look closely at the summary:

“So, now you owe each couple of grandparents \$999, together you owe them \$1,998. And you have \$1 in your hands. As you started with \$2,000, this means that \$1 + \$1,998 = \$2,000.”

\$1998 represents the total amount still owed. \$1 is already counted in \$1998, it’s just the money that you happen to have in hand at the moment, but it is not proper to add it to the owed amount again because then it would be counting it twice. The formula that would make sense would be: 2 + 1998 = 2000, where 2 represents the 2 dollars already given back and 1998 still owed.

Wang said...

The math isn't quite right since you're adding up different quantities:

Your grand parents each give you \$1000 so you owe them \$2000.

You get \$3 in change from a cost of \$1997.

You give each of your grand parents \$1 so you have \$1 left. Since you gave them \$2 (in total), then you owe them \$1998 (in total) and that adds up to \$2000. You can't just add up what you owe them and what yo have left to get the total amount.

As you said, this is similar to another famous missing dollar puzzle. If I recall correctly, that one involves three businessmen at a hotel.

In our case, we are just looking at the cashflow wrong. We started with \$2,000, returned a dollar to each grandparent, kept one and spent the remaining \$1,997 for camp. An equation representing this \$2000 - 2(\$1) -\$1 = \$1,997.

Jerome said...

You cannot mix what you owe with what you did with the money.

Here's what you owe.

Maternal Grand Parents donated 1000
Balance owed 999
What you paid 1
====================================
Total 1000 1000

Paternal Grand Parents looks the same way.

The dispersion is quite different

Given by both sets of grandparents 2000
Fees 1997
Retirm tp PGP 1
Kept for self 1
========================
Total 2000 2000

Moral: You cannot mix and match the dispersion with what you owe. All sorts of things can happen when you disobey accounting rules. They are in place to prevent this problem from occuring.

Annie said...

I think the flaw is when you assume you owe them together \$1998. The \$1. you kept was taken out of the equation. You actually owe them \$1999 or \$999.50 each. You owed them each \$998.50 for the tuition. If you gave them back \$1., then you owe them each \$999.50, the extra \$.50/each coming from the \$!. that you kept.

Anonymous said...

You start with \$2000 with the camp costing \$1997 and should get \$3 back. \$1997 + \$ 1 for mom's parents and \$1 for dad's parents equal \$1999 leaves \$1 for the camper. With this proposition, you are adding the one you get back to the cost of the camp to get \$1998 and then saying where is the rest of the money without giving money to the grandparents.

Gurubandhu

Maria said...

You are all right - we can't add apples to oranges. This is a logical confusion that can be easily seen if we assume that camp costs \$1,995 instead of \$1,997.
Then you get \$2,000 from the grandparents, receive \$5 change that you split between the grandparents. Each grandparent couple therefore gave you \$1,000 - \$2 = \$998
You owe them \$1,996 and all you have is a \$1.
If you give them this \$1, you will owe them \$1,995 that is exactly the cost of the camp.

A puzzle point for everyone who answered.