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Packing Puzzles

How many balls can you fit in one sphere?


What are Packing Puzzles

Packing puzzles are part of what is known as recreational math, but they are also called optimization problems. Indeed, some regular people might consider this types of problems fun, but for manufacturers they can bea real problem. A company that produces tennis balls actually needs a solution to fit as many as possible in one container to cut costs. Did you know that a Japanese company once invented square watermelons, not as a novelty to impress the public with, but to solve a simple packing problem. (The public did not appreciate the odd-looking melons and the idea was dropped!)

All packing puzzles have basically the same goal, finding a way to fit as many identical items in as few containers as possible.


Types of Packing Puzzles

Puzzles in two dimensions are just for fun as they have no practical utility. One example refers to packing sand, specifically grains of sand wasting as little space as possible. The grains are represented as circles and the idea is to find a compact way of stacking them.

Most packing puzzles in three dimensions concern spheres, as cubes are way easier to stack, not much of a puzzle there, you only need basic geometry.

Since most real-life packaging problems revolve around boxes, many puzzles ask the players to find the highest number of spheres that can be crammed into a cuboid. Unlike many other puzzles that require logical deductions or lateral thinking, to solve packing puzzles you have to use what you learned in school.


Famous Packing Puzzles

One of the best-known packing puzzle is the so-called SlothouberGraatsma puzzle which challenges the player to pack six 1 x 2 x 2 blocks and three 1 x 1 x 1 blocks into a 3 x 3 x 3 box. Unlike other puzzle, this one has only one solution.

Another example is the Conway puzzle, in which you have to pack rectangular blocks of various dimensions into a cubic


The Knapsack problem is a more complex packing puzzle as it adds another element, which is value. What you have is a set of items with different weights and value. The goal of the puzzle is to determine the number of items to put in a knapsack so that the total weight is less than or equal to a given limit and the total value is as large as possible.