PUZZLE 1 Handshake Problem
I invite ten couples to a party at my house. I ask everyone present, including my wife, how many people they shook hands with. It turns out that everyone questioned — I didn’t question myself, of course — shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my wife shake hands with? (I did not ask myself any questions.)
PUZZLE 2 Strawberry Ice Cream
I visited a math professor of mine for dinner at his home (well, not really but shh! it’s part of the problem!) who had pictures of his three daughters on his mantle. He had had pictures taken of the three girls when each was a particularly adorable age — the same age for all three, as it happens. Unfortunately, this made it impossible for me to determine which was the oldest. So I had to ask him. Being a math professor, however, he declined to answer directly, telling me only that the product of their current ages was 72. “However,” he added, “since that isn’t enough information to determine their ages, I’ll also tell you that the sum of their ages happens also to be the number of our street address.”
(Of course, I understood that each daughter’s age was to be considered an integer for this puzzle.)
I darted outside to check the number on his mailbox. I was daunted to discover that I still didn’t have enough information to determine their ages, and I returned to tell him so. “That is an astute observation,” he said, smiling. “So you’ll be glad to know that my oldest daughter prefers strawberry ice cream.”
Finally! I knew their ages. Do you ?
PUZZLE 3 Dissection Dilemma
The top two figures show how each of two shapes can be divided into four parts, all exactly alike. Your task is to divide the blank square into five parts, all identical in size and shape
PUZZLE 4 100 Light Switches
I give you a row of 100 light switches, all in the off position. Starting from the left, I ask you to flip every switch. Again starting from the left, I ask you to flip every other switch — so flip the 2nd, the 4th, etc. Again starting from the left, please flip every third switch. And so on: every fourth, then every fifth, etc, until on the last pass you flip only “every hundredth switch,” which means only the rightmost switch. When we are finished, which light switches are in the on position, and which are in the off position ?
PUZZLE 5 George’s Ropes
George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.
PUZZLE 6 Are You Sure There’s No Typo ? Find the missing number in this sequence:
I invite ten couples to a party at my house. I ask everyone present, including my wife, how many people they shook hands with. It turns out that everyone questioned — I didn’t question myself, of course — shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my wife shake hands with? (I did not ask myself any questions.)
PUZZLE 2 Strawberry Ice Cream
I visited a math professor of mine for dinner at his home (well, not really but shh! it’s part of the problem!) who had pictures of his three daughters on his mantle. He had had pictures taken of the three girls when each was a particularly adorable age — the same age for all three, as it happens. Unfortunately, this made it impossible for me to determine which was the oldest. So I had to ask him. Being a math professor, however, he declined to answer directly, telling me only that the product of their current ages was 72. “However,” he added, “since that isn’t enough information to determine their ages, I’ll also tell you that the sum of their ages happens also to be the number of our street address.”
(Of course, I understood that each daughter’s age was to be considered an integer for this puzzle.)
I darted outside to check the number on his mailbox. I was daunted to discover that I still didn’t have enough information to determine their ages, and I returned to tell him so. “That is an astute observation,” he said, smiling. “So you’ll be glad to know that my oldest daughter prefers strawberry ice cream.”
Finally! I knew their ages. Do you ?
PUZZLE 3 Dissection Dilemma
The top two figures show how each of two shapes can be divided into four parts, all exactly alike. Your task is to divide the blank square into five parts, all identical in size and shape
PUZZLE 4 100 Light Switches
I give you a row of 100 light switches, all in the off position. Starting from the left, I ask you to flip every switch. Again starting from the left, I ask you to flip every other switch — so flip the 2nd, the 4th, etc. Again starting from the left, please flip every third switch. And so on: every fourth, then every fifth, etc, until on the last pass you flip only “every hundredth switch,” which means only the rightmost switch. When we are finished, which light switches are in the on position, and which are in the off position ?
PUZZLE 5 George’s Ropes
George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.
PUZZLE 6 Are You Sure There’s No Typo ? Find the missing number in this sequence:
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