Did you solve it? Fields Medals for Beginners | Math

Earlier today, I posed three problems to you, inspired by the 2022 Fields Medals. The awards – which recognize up to four mathematicians under the age of 40 every four years – are the most famous prizes in mathematics.

Maryna Viazovska from Ukraine won the award for her groundbreaking work on how to wrap 24-dimensional spheres. The first puzzle was how to wrap beers in three dimensions.

1. A checkout problem

Is it possible to put more than 40 cans of beer with a diameter of 1 unit and a height of 2.6 units in a box with dimensions 5 x 8 x 2.6?

Here are 40 in the crate. But can you fit in more?

The solution Yes it’s possible. You can get 41 of them by packing in a hexagonal way

beer puzzle

If you don’t have enough cans at home to prove it, Pythagoras can prove it for you. The Pythagorean theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So in the triangle below, 12 = X2 + (0.5)2Where X = √(0.75) = 0.87 to two decimal places.

beer puzzle
The vertices of this triangle are the center of two cans and the point where two cans meet. The horizontal and vertical lines meet at right angles. If the diameter is 1 unit, the radius is 0.5 units.

When nine vertical rows (of 41 cans) are stacked, the horizontal distance is 0.5 + 8X + 0.5 = 1 + (8 x 0.87) = 1 + 6.93 = 7.93. This number is less than 8, so we know the cans will fit.

The second puzzle, on the prime number 13, was inspired by Briton James Maynard’s Fields Medal for his many prime results on prime numbers.

2. Chairs, man.

Place 13 chairs along the walls of a rectangular room so that each wall has the same number of chairs as the wall it faces.

The solution

chair puzzle
One of the chairs is in a corner. The others are on the sides.

The third puzzle was a tribute to June Huh, whose Fields Medal was awarded for results linking graph theory, combinatorics, algebra, and many other abstract concepts. A graph in this context means a network of discrete dots connected to each other, which is a way of thinking of a chessboard, which is made up of discrete squares connected to each other.

3. Chess Neighbors

Imagine a 9×9 chessboard. (Like a Sudoku grid, but with alternating black and white cells). Each square has a different person standing on it. Is it possible for all 81 people to walk to a neighboring square, so that each square again has a different person on it?

The solution No.

If everyone is on a square, that means 40 people are on one color of square, and 41 are on the other color of square. When everyone moves to a neighboring square, all black people go white, and vice versa. But this move is impossible because the number of black squares is not equal to the number of white squares.

I hope you enjoyed today’s puzzles. I will be back in two weeks.

PS Don’t forget to buy the Guardian on Saturday (July 16). I have edited a 16 page summer puzzle supplement which will be free with the physical paper. The supplement features puzzles from around the world, including hand-crafted sudoku by our friends at Cracking the Cryptic, all-new Japanese logic grid puzzles, a selection of Grabarchuk family teasers, many types of word puzzles and several crossword puzzles, including a guide to how to solve the puzzles. Don’t miss it!

Sources: 1. Trần Phương, 2&3 Half a Century of Pythagoras Magazine.

I install a puzzle here every two weeks on a Monday. I’m always on the lookout for great puzzles. If you want to suggest one, write to me.

I’m the author of several puzzle books, as well as the Football School children’s book series. The last payment, The greatest quiz book of all time, just got out.

I give school lectures on math and puzzles (online and in person). If your school is interested, please contact us.

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