I place $20 in a box.
So do you.
Now the box contains $40, and we both know it.
I sell the box to you for $30.
And we both walk away with a $10 profit.

Jay sent over a link this great problem. It’s from one of my favorite blogs Futility Closet. This is the perfect kind of problem to throw out in class…easy to remember and easy to think through in a few minutes. What’s lovely is that initially it seems so plausible and yet impossible that they both profit $10. It reminds me of a Nova show (embedded below) I was watching on neuroscientists studying magic. Most tricks rely on our visual system’s strong bias toward detecting and anticipating motion. I feel like this problem and ones like it trick our reason, possibly with the momentum of language? Whoa, that got deep! Better stop right there.

I just heard another good problem along these lines:

A woman comes in to buy a $30 pair of shoes. She wishes to pay with a $50 bill. The shoe salesman doesn’t have any change so he goes next door and gives his neighbor the $50 bill in exchange for smaller bills to make change. He then sells the shoes and gives the woman her $20 in change. An hour later the neighbor comes to the shoe salesman and points out that the $50 bill is a counterfeit. The shoe salesman immediately returns his neighbors $50 in change.

How much has the shoe salesman lost in total money and in merchandise?

When I tried this problem I came up with $50. I tried to group every transaction by person and amount. So:
1st: Salesman and neighbor exchange fake $50
Salesman = +50 Neighbor = -50

2nd: Woman and salesman exchange the shoes (30) and change (20)
Saleman = 50 – 50 = 0 Woman = 30 + 20 = +50

I’m with you, Nick. I was doing the difference from his highest to lowest. Not from where he started to where he ended. Glad I checked that with you before using it in class!

I just heard another good problem along these lines:

A woman comes in to buy a $30 pair of shoes. She wishes to pay with a $50 bill. The shoe salesman doesn’t have any change so he goes next door and gives his neighbor the $50 bill in exchange for smaller bills to make change. He then sells the shoes and gives the woman her $20 in change. An hour later the neighbor comes to the shoe salesman and points out that the $50 bill is a counterfeit. The shoe salesman immediately returns his neighbors $50 in change.

How much has the shoe salesman lost in total money and in merchandise?

I got that he’s out $100. Good one. Who gave that one to you? Argazzi?

When I tried this problem I came up with $50. I tried to group every transaction by person and amount. So:

1st: Salesman and neighbor exchange fake $50

Salesman = +50 Neighbor = -50

2nd: Woman and salesman exchange the shoes (30) and change (20)

Saleman = 50 – 50 = 0 Woman = 30 + 20 = +50

3rd: Neighbor and salesman exchange fake $50 again

Salesman = 0 – 50 = -50 Neighbor = -50 + 50 = 0

When I do the problem this way I get the salesman has lost $50, the neighbor broke even, and the woman walked away with $50 in shoes and cash.

I’m with you, Nick. I was doing the difference from his highest to lowest. Not from where he started to where he ended. Glad I checked that with you before using it in class!