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### Discrete Reasoning Puzzles

These puzzles require sharp, logical thinking. Can you solve them?

## #31

Abel, Mabel, and Caleb went bird watching. Each of them saw one bird that none of the others did. Each pair saw one bird that the third did not. And one bird was seen by all three. Of the birds Abel saw, two were yellow. Of the birds Mabel saw, three were yellow. Of the birds Caleb saw, four were yellow. How many yellow birds were seen in all? How many non-yellow birds were seen in all?

Solution

## #32

Four gentlemen (Adam, Bill, Chuck, and Dan) went to an expensive restaurant to dine. They checked their coats, hats, gloves, and canes at the door (each of the gentlemen had one of each). But when they checked out, there was a mix up, and each of the men ended up with exactly one article of clothing (a pair of gloves is considered a single article of clothing) belonging to each one of the four. Adam and Bill ended up with their own coats, Chuck ended up with his own hat, and Dan ended up with his own gloves. Adam did not end up with Chuck's cane. State whose coat, hat, gloves, and cane each of the gentlemen ended up with.

Solution

## #33

You have ten boxes, each of which contains nine balls. The balls in one box each weigh 0.9 pounds; the balls in all the other boxes weigh exactly one pound each. You have an accurate scale in front of you, with which you can determine the exact weight, in pounds, of any given set of balls. How can you determine which of the ten boxes contains the lighter balls with only one weighing?

Solution

## #34

A card-shuffling machine always rearranges cards in the same way relative to the original order of the cards. All of the hearts, arranged in order from ace to king, were put into the machine. The cards were shuffled and then put into the machine again. After this second shuffling, the cards were in the following order: 10, 9, Q, 8, K, 3, 4, A, 5, J, 6, 2, 7. What order were the cards in after the first shuffle?

Solution

## #35

A father is four times as old as his son. In twenty years, he'll be twice as old. How old are they now?

Solution

## #36

Four intellectuals are lined up so that each intellectual can see the ones in front of him but not the ones behind him. (The back one can see the other three, and the front one can't see anybody.) One hat is placed on the heads of each of the intellectuals. (None of them may see the color of their own hat, but each may see the color of the hats on the intellectuals in front of him.) Each of the four hats are one of three different colors (red, white, and blue), and there is at least one hat of each color (so there's one duplicate). Each of the intellectuals, starting with the back and ending with the front, is asked the color of the hat he is wearing. Each of the intellectuals is able to deduce and give a correct answer out loud, in turn. What arrangement of the hats permits this to be possible without guessing (since the specific colors chosen are arbitrary, just indicate which two intellectuals must be wearing hats of the same color), and how did they do it?

Solution

## #37

There are four men (call them 1, 2, 3, and 4) standing in front of a firing squad in a line. They are all facing the same direction such that 1 is at the back of the line, and 4 is at the front. 1 and 3 are wearing black hats, and 2 and 4 are wearing white hats. Between 3 and 4 is a brick wall. So 1, at the back of the line, can see 2 and 3. 2 can see 3. Neither 3 and 4 can see anybody. The men know that two of them are wearing black hats and two of them are wearing white hats. The commander of the firing squad offers a challenge. The challenge is that he will let all of them go if only one of them correctly names the color of his own hat. The men are not allowed to talk amongst themselves. Which of the men know for sure the color of his hat?

Solution

## #38

How can you build pig pens so you can put nine pigs in four pens such that each pen has an odd number of pigs?

Solution

## #39

Of three men, one always tells the truth, one always tells lies, and one answers "yes" or "no" randomly. Each man knows which one each of the others are. You may ask three yes/no questions, each of which may only be answered by one of the three men, after which you must be able to identify which man is which. How can you do it?

Solution

## #40

An explorer was trekking through a remote jungle when he was captured by logic-loving cannibals. He was brought before the chief and told, "You may now speak your last words. If your statement is true, then we will burn you at the stake. If your statement is false, we will boil you in oil." The man thought for a moment, then made his statement. Perplexed, the clever cannibals realized they could do nothing but let him go. What did the explorer tell them?

Solution