Can You Solve The Footballers Boot Boxes Riddle?
Photograph of each respective boot. Credit Pro Soccer Direct. Image only used for illustration purposes for this riddle-posting.
Here’s a riddle. Ten outfield football players have their respective football boots placed in randomly in in boxes. Each players gets five tries at opening the boxes, trying to find their own branded football boots. (Thus a 50% chance of each individual finding their football boots.)
They’re not allowed to communicate about what they find. If the entire team fails to find their football boots, they’re all can’t play the big game tonight. And the odds of them all finding their instruments via random guessing is 1 in 1,024. But the left back has an idea that will radically increase their odds of success. What’s that big idea?
Rules.
1. Each football boots have been randomly placed in 10 boxes.
2. The pictures on the boxes don’t necessarily correspond to the football boot inside.
3. Each outfield footballer can open up to 5 boxes. They have to close all of the boxes they open.
4. All 10 outfield footballers must find their own football boots.
5. The outfield players can’t in anyway communicate to each other what they find. They can see what goes on from a visible distance.
Given all this, it seems rather hopeless. But think on it. What could you do if you talk it through beforehand, as this puzzle allows?
Answer and solution below.
Have you had a good think about the riddle? Well, here’s the solution to the riddle. At first look, it feels impossible to figure out. However, like any good rollercoaster, it all comes down to loops. Remember that, loops.
The strategy that the left back propose that. Everyone opens the box with the picture of your respective football boot. If your boot is inside, done. Otherwise look at whatever is in there, and then proceed to open the box with that picture on it. Keep going until you find your respective football boot. The rest of the team is skeptical, but amazingly enough, they all find what they need. Few hours later get to play football.
So why did the left back’s strategy work? Each outfield footballer follows a linked sequence that starts with the box whose outside picture matches their respective football boots and ends up with the box that actually containing it. Note that if they kept going, that would lead them back to the start, so this is a loop.
For example, if the boxes arranged like so,
-The player with the PUMA Future would open the first box (Box 8) with the picture of the PUMA Future to find the NIKE Tiempo (Box 4).
- Then go to the NIKE Tiempo box to find the ADIDAS X (Box 7).
- Proceeds to the ADIDAS X box to find UMBRO boots.
-And find his or her football boots, the PUMA Future in the UMBRO boot box (Box 6).
So the next player, who owns the NIKE Tiempo knows where to go, when it is his or hers turn. To the PUMA Future box. Same goes to the person who owns the ADIDAS X and the UMBRO boots.
Because by starting with the box with the picture of their respective boots. Each outfield players restricts their search to the loop that contains their boots. And there’s a decent odds of 35%. That all of the loops of length of five or less.
How do we calculate those odds? For the sake of simplicity, we'll demonstrate with a simplified case, four instruments and no more than two guesses allowed for each musician.
Let's start by finding the odds of failure, the chance that someone will need to open three or four boxes before they find their instrument. There are six distinct four-box loops. One fun way to count them is to make a square, put an instrument at each corner, and draw the diagonals. See how many unique loops you can find, and keep in mind that these two are considered the same, they just start at different points. These two, however, are different.
We can visualise the eight distinct three-box loops using triangles. You'll find four possible triangles depending on which instrument you leave out, and two distinct paths on each.
Another example
-The player with the NIKE Hypervenom would open the first box (Box 9) with the picture of the NIKE Hypervenom to find the PUMA One (Box 3).
-Then go to the PUMA One box to find the ADIDAS Predator (Box 10).
- Proceeds to the ADIDAS Predator box to find NIKE Mercurial Vapor.
-And find his or her football boots, the NIKE Hypervenom in the NIKE Mercurial vapor box (Box 5).
So of the 24 possible combinations of boxes, there are 14 that lead to failure, and ten that result in success. That computational strategy works for any even number of players, but if you want a shortcut, it generalises to a handy equation. Plug in ten outfield footballers, and we get odds of about 35%.
What if there were 1,000 footballers ? 1,000,000? As it increases, the odds approach about 30%.Not a guarantee, but with a bit of musician's luck, it's far from hopeless.