Archive for the ‘Puzzles’ Category.

Continue the Sequence: 742, …

This is the sequence of numbers n such that 3 times the reversal of n plus 1 is the number itself. In other words, n = 3*reversal(n)+1. For example, 742 = 3*247+1. In fact, 742 is the smallest number with this property. How does this sequence continue, and why?

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I am Hooked on Star Battles

I recently published Sergei Bernstein’s awesome Star Battle called Swiss Cheese. Another lovely Star Battle from him is called Hooks. You can play it online at puzz.link.

Sergei's Star Battle

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Swiss Cheese Star Battle

Star Battle is one of my favorite puzzle types. The rules are simple: put two stars in each row, column, and bold region (one star per cell). In addition, stars cannot be neighbors, even diagonally.

My son, Sergei Bernstein, recently designed a Star Battle with a beautiful solve path. This is my favorite Star Battle so far. I like its title too: Swiss Cheese.

Swiss Cheese Star Battle

You can also solve it at the puzz.link Star Battle player.


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A Splashy Math Problem

A problem from the 2021 Moscow Math Olympiad went viral on Russian math channels. The author is Dmitry Krekov.

Problem. Does there exist a number A so that for any natural number n, there exists a square of a natural number that differs from the ceiling of An by 2?

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The Anniversary Coin

Konstantin Knop, the world’s top authority on coin-weighing puzzles, suggested the following problem for the 2019 Russian Math Olympiad.

Puzzle. Eight out of sixteen coins are heavier than the rest and weigh 11 grams each. The other eight coins weigh 10 grams each. We do not know which coin is which, but one coin is conspicuously marked as an “Anniversary” coin. Can you figure out whether the Anniversary coin is heavier or lighter using a balance scale at most three times?

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Discover the Rule

Found the following cute puzzle on Facebook.

Puzzle. Discover the rule governing the following sequence to find the next term of the sequence: 8, 3, 4, 9, 3, 9, 8, 2, 4, 3.

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Build an All-red Cube

This is one of my favorite problems given at the 2017 Moscow Olympiad to grades 6 and 7. It was suggested by one of my favorite problem writers: Alexander Shapovalov.

Problem. We are given eight unit cubes. The third of the total number of their faces are blue, and the rest are red. We build a large cube out of these cubes so that exactly the third of the unit cube’s visible faces are red. Prove that you can use these cubes to build a large cube whose faces are entirely red.

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2020 MIT Mystery Hunt

Every year I write about latest MIT Mystery Hunt puzzles that might be appealing to mathematicians. Before diving into mathy puzzles, I would like to mention two special ones:

Unfortunately math wasn’t prominent this year:

  • Food Court—This is a probability puzzle that is surprisingly uninspiring. There is no mystery: the puzzle page contains a list of probability problems of several famous types. But this puzzles can find great use in probability classes.
  • Torsion Twirl—Mixture of dancing and equations. I love it.
  • People Mover—Logical deduction at the first stage.

On the other hand, Nikoli-type puzzles were represented very well:

  • The Ferris of Them All—Several different Nikoli puzzles on a wheel.
  • Toddler Tilt—Not exactly a Nicoli puzzle, but some weird logic on a grid, some music too.
  • The Dollhouse Tour—Not exactly a Nicoli puzzle, but some weird logic on a grid, some pictures too.
  • The Nauseator—The first part of the puzzle is a huge nonogram.
  • Domino Maze—A non-trivial Thinkfun puzzle.
  • Backlot—Finding a path on a grid with a fractal structure.
  • Whale—Variation on Rush Hour.

Some computer sciency puzzles:

Cryptography:

A couple of puzzles with the mathy side hidden:

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SET Tic-Tac-Toe

The academic year is over and my junior PRIMES STEP group finished their paper about a classification of magic SET squares. A magic SET square is a 3 by 3 square of SET cards such that each row, column, and diagonal is a set. See an example below. The paper is posted at the arXiv:2006.04764.

A magic SET square

In addition to classifying the magic SET squares, my students invented the game of SET tic-tac-toe. It is played on nine cards that form a magic SET square. Two players take turns picking a card from the square. The first player who has a set wins.

One might think that this game is the same as tic-tac-toe, as a player wins as soon at they have cards from the same row, column, or diagonal. But if you build a magic SET square, you might notices that each magic SET square contains 12 sets. In addition to rows, columns, and diagonals, there are sets that form broken diagonals. The picture below shows all the sets in a magic SET square.

Sets in magic SET squares

There are more ways to win in this game than in a regular tic-tac-toe game. My students proved that ties are impossible in this game. They also showed, that, if played correctly, the first player always wins.

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A Game with the Devil

My former student, Dai Yang, sent me the following cute puzzle:

Puzzle. You are playing a game with the Devil. There are n coins in a line, each showing either H (heads) or T (tails). Whenever the rightmost coin is H, you decide its new orientation and move it to the leftmost position. Whenever the rightmost coin is T, the Devil decides its new orientation and moves it to the leftmost position. This process repeats until all coins face the same way, at which point you win. What’s the winning strategy?

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