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# Problem of the Month Archive

## January 2012

### 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

## February 2012

### Lots of 7's

What is the last digit of $$77^{77}$$?

Solution

### Even More 7's

What is the last digit of $$77^{77^{77}}$$?

Solution

## March 2012

### 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

## April 2012

### 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

## May 2012

Beheaded is a solitaire game played with a row of pennies. To play, arrange some pennies in a row. A move consists of removing a penny that is heads up. If there was a penny immediately to the left of the removed penny, it is flipped over. Likewise, if there was a penny immediately to the right of the removed penny, it too is flipped over. To win the game, you must, through a series of moves, remove all the pennies.

For example, in the row of pennies to the right, for your first move you remove the circled penny, and that causes the two pennies adjacent to it to be flipped over. In the second move, you remove the cirlced penny. Only one penny is immediately adjacent to the removed penny and it is flipped over. At this point since no pennies are heads up, you can no longer move and the game ends in a loss.

Is it possible to win beheaded with the row of pennies below. Show how, or explain why it is not possible to win

Solution

Is it possible to win beheaded with the row of pennies below. Show how, or explain why it is not possible to win

Solution

## June-July 2012

Angie, Bonnie, and Carly toss coins. Angie tosses her coin 9 times, Bonnie tosses hers 10 times, and Carly hers 11 times. Which of the three girls is most likely to toss fewer heads than tails?

Solution

Angie, Bonnie, and Carly toss their coins again. This time Angie tosses hers 10 times, Bonnie 11 times, and Carly 12 times. Which of the three girls is most likely to toss fewer heads than tails?

Solution

## August 2012

### 20 Divisors

What is the smallest number with exactly 20 divisors? (For example, 10 has 4 divisors: 1, 2, 5, 10.)

Solution

### 20 Divisors Again

What is the smallest number divisible by 7 with exactly 20 divisors? (For example, 10 has 4 divisors: 1, 2, 5, 10.)

Solution

## September 2012

### 2x4 Cards

Arrange the 8 cards in a row so that between the two aces there is one card, between the two twos is two cards, between the two threes is three cards, and between the two fours is four cards.

Solution

### 4x4 Cards

Arrange the 16 cards into a square of four rows of four cards each, so that in each row and column each rank appears exactly once and each suit appears exactly once.

Solution