**Puzzle 31**

Starting with the number 12.5, you may add, subtract, multiply by or divide by any one-digit number, and with the result, you may again do any one of those operations with any one-digit number. Which operation with what one-digit number at each stage will produce the number 38?

Show Solution

**Puzzle 32**

Find two digits to replace “A” and “B” in the number A123456789B that will make the resulting 11-digit number divisible by 36. There are two solutions.

Show Solution*xy*if and only if it is divisible by

*x*and by

*y*. Let’s express 36 as 4•9, as there is a test for divisibility by each of these factors. For a number to be divisible by 4, its rightmost two digits must form a 2-digit number that is divisible by 4. So B = 2 or B = 6, as both 92 and 96 (and no other numbers between 90 and 99) are divisible by 4. For a number to be divisible by 9, the sum of its digits must be divisible by 9. Since the sum of the digits 1 through 9 is 45, which is divisible by 9, for the sum of all 11 digits to be divisible by 9, A + B must be divisible by 9, so if B is 2, A is 7, and if B is 6, A is 3. You can use your calculator to confirm that both of the two resulting numbers, 71234567892 and 31234567896, are, in fact, divisible by 36.

**Puzzle 33**

The interviewer from Puzzle 27 found himself in the same fix a year later and this time he said to the three perfect logicians applying for the logician position, “As you can see, there are ten white hats and ten black hats in this closet. I will blindfold all of you and put one of these hats on each of you.” He then blindfolded them, placed a white hat on each of them, closed the closet door, removed their blindfolds and said, “If you see anyone wearing a white hat, raise your hand.” Naturally, all three applicants raised their hands. He then said, “The first one of you to deduce what color hat he is wearing gets the job.” A few second later, a second before the other two applicants nailed it, too, one of them said. “I’m wearing a white hat” and logically justified his conclusion. How did he know?

Show Solution*not*be raising his hand, so I must be wearing a white hat’, but B hasn’t said anything, so I must be wearing a white hat.”

**Puzzle 34**

If it takes ten minutes for ten sheep to jump consecutively over a four-foot-tall fence, how many sheep can jump over that fence in 60 minutes? Assume that the timing begins at the beginning of the first sheep’s jump and ends at the end of the last sheep’s jump.

Show SolutionThis puzzle is analogous to one that says, “If 10 vertical posts are required to build a 10-foot-long fence, how many posts are required to build a 60-foot-long fence?” If the distance between the first post and the tenth post is 10 feet, then, since there are 9 spaces between those two posts, the distance between any two consecutive posts is 10/9 of a foot. Dividing 60 feet by 10/9 of a foot per post = 54 posts, plus the first post = 55 posts.

It takes only a second to jump four feet up and return to the ground, so the ten minutes that it takes ten sheep to jump consecutively over the four-foot-tall fence must be the sum of the nine periods of time between jumps, so the time period from the end of one jump to the end of the next is 10/9 of a minute. Dividing 60 minutes by 10/9 of a minute per sheep = 54 sheep, plus the first sheep = 55 sheep.

**Puzzle 35**

See diagram below. A right angle is drawn so that both sides are tangent to a circle whose radius measures 1. In the space bounded by the circle and the right angle, a smaller circle is drawn tangent to the larger circle and to both sides of the right angle. What is the measure of the radius of the smaller circle?

*x*= the radius of the smaller circle, then each leg of the right triangle measures 1 –

*x*and the hypotenuse measures 1 +

*x*. Therefore,