Problem of the Month Archive

Turning 2012

Imagine an array of red dots extending forever in all directions. A path starts at one dot and moves outward in a spiral manner as shown in the diagram to the right (the starting dot is outlined in blue). After passing through the 2012th dot, how many turns has the path made?

Solution

2012 Turns

As you move along the path as shown, number the dots sequentially starting at 1. What is the number of the dot on which the 2012th turn is made?

Solution

Lots of 7's

What is the last digit of \(77^{77}\)?

Solution

Bills and Hats I

"No," said the mathematician to his 12-year old son, "I do not feel inclined to increase your allowance this week from $3 to $5. But if you'll take a risk, I'll make you a sporting proposition.

"I'll use two hats and in the first hat I'll place five $5 bills and five $1 bills, while in the second I'll put one $5 bill and four $1 bills. You may select either of the hats. If you select the first hat, you may without looking choose one bill from the hat, while if you select the second hat, you may without looking choose two bills from it."

Should the boy accept the offer? If so, which hat should the boy choose to pick bills from in order to maximize his allowance?

Solution

Bills and Hats II

After a few more weeks, the son said, "I really feel like I should have a better chance to receive a higher allowance. In fact I think my allowance ought to be $5 more a week!"

"Whoa," said the mathematician to his son, "I do not feel inclined to increase your allowance this week by $5. But here's my new proposal:

"I will use five $5 bills and five $1 bills. You may divide them any way you please into two sets. We'll put one set into one hat, the other set into a second hat. I'll mix up the hats and put one hat on the right and one on the left. You pick either hat at random, then reach into that hat and choose one bill. Whatever you choose will be your allowance."

Should the boy agree to this proposal? If so, how should he divide the ten bills between the two hats in order to maximize his allowance?

Solution

11 Gold Coins

You are given 12 identical looking gold coins. 11 are solid gold but one is fake. The gold coins are all equal in weight, while the fake one is lighter than the rest. Using a balance, what is the greatest number of genuine gold coins that you can identify in two weighings?

Solution

10 Gold Coins

You are given 12 identical looking gold coins. 10 are solid gold but two are fake. The gold coins are all equal in weight, while the fake ones are equal in weight but lighter than the solid gold ones. Using a balance, what is the greatest number of genuine gold coins that you can identify in two weighings?

Solution