For those who love a mental workout, a set of three deceptively simple geometry puzzles recently challenged readers to think outside the box. The puzzles, sourced from mathematician Ian Stewart, test logic, spatial reasoning, and a dash of creative thinking. Below, we explore the intriguing problems and reveal their clever solutions.
The Impossible Tile Grid
The first conundrum, dubbed 'Bonnie Tiler', presented a square grid missing three corner cells, leaving 33 cells in total. The task was to cover this irregular grid using 11 identical tiles, each made of three cells in a straight line. At first glance, it seems perfectly plausible.
However, the solution reveals a cunning impossibility. By colouring the grid in a repeating pattern of three colours—blue, yellow, and red—a critical property emerges. Every possible placement of the three-cell tile covers one cell of each colour. Therefore, for a complete covering to exist, the grid must contain an equal number of each coloured cell.
A careful count shows the grid has 12 red cells and only 10 yellow cells, making a perfect cover with the 11 tiles mathematically impossible. The colouring trick elegantly proves why no solution can be found.
Cutting and Assembling a Square
The second puzzle, 'Assembly Needed', asked for ingenuity. A given shape, composed of smaller squares, can be cut along its black lines into four identical pieces that reassemble into a perfect square. One solution was provided, but the challenge was to find a different way to achieve the same result.
The alternative solution requires seeing the shape not as a whole, but as a set of modular, interlocking parts. By cutting the left-hand shape differently—still into four congruent pieces—and then rotating and reflecting them, a new configuration forms the required square. This puzzle highlights the fascinating area of dissection problems, where visualising transformations is key.
The Optimal Pizza Division
The final, mouth-watering problem was the 'Pizza Party'. With three identical pizzas to share among five people, the goal was to find the method requiring the smallest number of total pieces, while ensuring everyone gets exactly the same amount of pizza in terms of both the number and size of their slices.
One method is to cut each pizza into five equal slices, giving 15 pieces in total, with each person receiving three slices. Another involves giving three people a 3/5 slice and two people a combination of a 2/5 and a 1/5 slice.
The most efficient solution, however, requires just ten pieces in total. The method is elegantly simple: cut two of the pizzas in half, creating four halves. Cut the remaining pizza into ten equal slices. Each person then receives one half-slice and one tenth-slice, perfectly equating to 3/5 of a pizza each. This minimises cuts and ensures absolute fairness.
These puzzles, conceived by Ian Stewart, offer a brilliant demonstration of how mathematical principles underpin seemingly simple games. Stewart's new book, Reaching for the Extreme, is available for pre-order ahead of its release on 12 February.
