Archive for the ‘Puzzles’ Category.

## 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?

Share:## 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.

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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?

## 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?

## Puzzle Ninja

Alex Bellos sent me his new book Puzzle Ninja: Pit Your Wits Against The Japanese Puzzle Masters. What has he done to me? I opened the book and couldn’t close it until I solved all the puzzles.

This is a fantastic book. There are many varieties of puzzles, including some types that I’ve never seen before. Also, the beautifully designed puzzles are great. Often puzzles of the same type target different solving ideas or have varied cool themes.

This book is more than a bunch of puzzles; it also contains poetic stories about puzzle histories and Japanese puzzle designers. Fantastic puzzles together with a human touch: this might be my favorite puzzle book.

I present two puzzles from the book. The puzzle type is called *Wolf and Sheep Slitherlink*.
The Slitherlink is a famous puzzle type with the goal of connecting
some of the neighboring dots into a single non-self-intersecting loop. A
number inside a small square cell indicates how many sides of the
square are part of the loop. *Wolf and Sheep Slitherlink* is a variation of *Slitherlink* in which all sheep should be kept inside the fence (loop) and all the wolves outside.

Ignore the numbers in the title as they just indicate the order number of Wolf and Sheep Slitherlink puzzles in the book. The number of ninja heads shows the level of difficulty. (The hardest puzzles in the book have four heads.) The difficulty is followed by the name of the puzzle master who designed the puzzle.

The first puzzle above is slightly easier than the second. I like the themes of these two puzzles. In the first one, only one cell—lonely wolf—marks the relationship to the fence. In the second one, the wolf in the center—who needs to be outside the fence—is surrounded by a circle of sheep who are in turn surrounded by a circle of wolves.

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## Mafia Logic Puzzle

I found this puzzle on the Russian QWERTY channel.

Five people sit around a table playing Mafia. Among them are two innocent people, two Mafiosos, and one detective. The Mafia people know each other; the detective knows who each of them is; and the innocent people have no information whatsoever about anyone at the table.

During this particular game, the innocents and the detective always tell the truth, while mafia people always lie. They start by going around the circle making the following statements:

- A: I know who B is.
- B: I know who the detective is.
- C: I know who B is.
- D: I know who E is.
Who is who?

## Another Cool Coin-Weighing Problem

My coauthor, Konstantin Knop, sent me a coin-weighing problem that is really good. Surprisingly, it is old: it first appeared in a Russian math journal, *Kvant*, in 1973.

Puzzle.At a trial, 14 coins were presented as material evidence. The expert tested the coins and discovered that seven of them were fake, the rest were real, and he knew exactly which coins were fake and which were real. The court only knows that counterfeit coins weigh the same, real coins weigh the same, and fake ones are lighter than real ones. The expert wants to use not more than three weighings on a balance scales without weights to prove to the court that all the counterfeit coins he found are really fake, and the rest are real. Could he do it?