## A Probability Puzzle

Puzzle. You got two envelopes with two distinct real numbers. You chose one of them and open it. After you see the number you are allowed to swap envelopes. You win if at the end you pick the larger number. Find a strategy that gives you a probability more than 1/2 of winning.

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## What are the Numbers?

Another cute puzzle found on Facebook.

Puzzle. A teacher wrote four positive numbers on the board and invited his students to calculate the product of any two. The students calculated only five of six products and these are the results: 2, 3, 4, 5, 6. What is the last product? What are the original four numbers?

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## Another Weird Test Question

I found this puzzle on Facebook:

Puzzle. Solve this:
1+4 = 5,
2+5 = 12,
3+6 = 21,
5+8 = ?
97% will fail this test.

Staring at this I decided on my answer. Then I looked at the comments: they were divided between 34 and 45 and didn’t contain the answer that initially came to my mind. The question to my readers is to explain the answers in the comments and suggest other ones. Can you guess what my answer was?

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## Day and Night

Puzzle. The length of the day today in Boston is the same as the length of the coming night tonight. What is the total length of both?

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## How Old is Everyone?

My friend Alice reminds me of me: she has two sons and she is never straight with her age. Or, maybe, she just isn’t very good with numbers.

Once I visited her family for dinner and asked her point blank, “How old are you?” Here is the rest of the conversation:

Alice: I am two times older than my younger son was 5 years ago.
Bob: My mom is 12 times older than my older brother.
Carl: My younger brother always multiplies every number he mentions by 24.
Bob: My older brother is 30 years older than me.
Carl: My mom is 8 times older than me.
Alice: My older son always multiplies every number he mentions by 2.

How old is everyone?

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## Cube Sudo-Kurve

Last year, when I read an application file of Wayne Zhao to PRIMES, I got very excited because he liked puzzles. And I’ve always wanted to have a project about puzzles. After Wayne was accepted to PRIMES we started working together. Wayne chose to focus on a variation of Sudoku called Sudo-Kurve.

We chose a particular shape of Sudo-Kurve for this project, which ended up being very rewarding. It is called Cube Sudo-Kurve. The Cube Sudo-Kurve consists of three square blocks. The gray bent lines indicate how rows and columns continue. For example, the first row of the top left block becomes the last column of the middle block and continues to the first row of the bottom right block. As usual each row, column, and square region has to have 9 distinct digits.

Wayne and I wrote a paper Mathematics of a Sudo-Kurve, which has been published at Recreational Mathematics Magazine.

A Cube Sudo-Kurve needs at least 8 clues to have a unique solution. Here we have a puzzle with 8 clues that we designed for our paper.

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## Emissary Puzzles

I’ve been invited to help with the Puzzle Column at the MSRI newsletter Emissary. We prepared six puzzles for the Fall 2018 issue.

I love the puzzles there. Number 2 is a mafia puzzle that I suggested. Number 6 is a fun variation on the hat puzzle I wrote a lot about. Here is puzzle Number 3.

Puzzle. Let A = {1,2,3,4,5} and let P be the set of all nonempty subsets of A. A function f from P to A is a “selector” function if f(B) is in B, and f(B union C) is either equal to f(B) or f(C). How many selector functions are there?

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## Cave Lioness

(Photo by Rebecca Frankel.)

When I was in grade school, one of the teachers called me Cave Lioness. She hated my unruly hair, which reminded her of a lion’s mane. This teacher was obviously very uninformed, for female lions do not have manes.

This name calling had the opposite to the desired effect. I became proud of my mane and didn’t ever want to cut it. When I grew older, I opted for convenience and started to cut my hair short&mdahs;sometimes very short.

Last year I was too busy for barbers, and my hair grew more than I intended. As it turned into a mane, I remembered the story of this nickname.

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## My Phone

Once I was giving a math lecture and my phone, which I’ve never quite understood, was on the desk in front of me. Suddenly it rang. I didn’t pick it up, as I proceeded with my lecture. The ringing stopped, while I was explaining a particularly interesting mathematical point. After a minute, my phone said, “I do not understand a word you are saying.”

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## The Halfsies

Detective Radstein is investigating a robbery. He apprehends three suspects: Anne, Bill, and Caroline. The detective knows that no one else could have participated in the robbery. During the interrogation the suspects make the following statements:

• Anne: I didn’t do it. Bill did it alone.
• Bill: I didn’t do it. Caroline did it.
• Caroline: I didn’t do it. Bill did it.

Detective Radstein also discovered that all three suspects are members of a club called The Halfsies. Every time they speak, they make two statements, one of which is a lie and the other one is true. Who committed the robbery?

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