A sheet of squared paper, two people. Each builds an invisible maze for the other.
In 2003 the Italian game writer Andrea Angiolino put it in Mind-Sharpening Logic Games. In 2022 Ben Orlin put it in the information games chapter of Math Games with Bad Drawings. Both books call it the Franco-Prussian Labyrinth.
You already know
Mazes. You trace a line from the way in to the way out, however much it twists.
The odd part
In this maze you cannot see the walls. You try one step at a time and only find a wall by hitting it. It looks like pure luck.
You will ask
Can anyone calculate walls they cannot see?
Learn to play. Three rules.
1
Build a maze for the other player: a 9×9 grid with 30 walls. Leave a path from the start to the finish.
2
Take turns walking through each other's maze. The walls are hidden. A turn is at most 5 steps, and hitting a wall ends it.
3
Walls you hit are drawn on your map. Whoever reaches the finish at the bottom right first wins.
What the mathematicians say
Walking a maze blind looks like luck, and every step can be counted. Count well, and you hit fewer walls and reach the finish sooner.
Why do you say that?Follow one game: build the maze, walk it in the dark, race to the finish.
OPENING
Draw the maze as dots and lines, and you know the most walls it can hold
Major premise
Make each cell a dot. Where two cells have no wall between them, join them with a line. Each new cell you join needs at least one more passage.
Minor premise
A 9×9 board has 81 cells. Between the cells there are 144 places a wall can go.
Conclusion
81 cells need at least 80 passages, and 144−80=64. With at most 64 walls every cell can still be reached. You only have 30, so the path cannot be shut.
MIDDLE
Every step asks the maze one question
Each step asks: is there a wall here? The maze only answers yes or no. One answer like that is what mathematicians call a bit.
If you get through, you know there is no wall. If you hit one, your turn ends, and you know there is a wall. Either way, one fewer place is unknown.
The walls you hit stay on your map, so next turn you know which way not to try. Walking back the way you came asks nothing new.
ENDING
At least 16 steps, and a detour only adds an even number
Major premise
A turn is at most 5 steps, so 4 turns are at most 20 steps.
Minor premise
The shortest path is 16 steps: 8 right and 8 down. A detour always adds an even number, so the next lengths are 18 and 20.
Conclusion
16, 18 and 20 steps all fit in 4 turns. A bend or two costs nothing. Hitting a wall and losing the rest of your steps is what hurts.
Opening How many walls at most? Try another maze and count
This one: 80 passages, 64 walls
Tap me
Drag me
Places for walls144
Passages needed80
Most walls64
144 − 80 = 64
Count the places for walls like this: each row has 8, and there are 9 rows. Each column has 8 too, and there are 9 columns. 9×8×2 = 144.
Line up the most walls for each board. What do you notice?
2×21
3×34Tap me
4×49
5×516
6×625
7×736
8×849
9×964
10×1081
1, 4, 9, 16, 25… each is a number times itself. A 2×2 board holds at most 1×1 walls, 3×3 holds 2×2, and 9×9 holds 8×8=64. When a maze has the most walls it can hold, there is exactly one path between any two cells. Mathematicians call a maze like that a tree.
Middle Every step leaves one fewer place unknown
Tap me
You stand on the start at the top left. You cannot see a single wall. There are 144 places between cells, and you know nothing about any of them.
144 places unknown
No wallWallUnknown
5 steps left this turn
Ending A detour always adds an even number of steps
Tap me
The shortest path: 8 right and 8 down, 16 steps in all.
16 steps
5551
5+5+5+1, 4 turns
The start and the finish are both white. The number in a cell says which step landed there: white cells hold even numbers, grey cells odd ones. Each step changes colour, so the finish always takes an even number of steps. 17 or 19 can never get there.
Walls you cannot see,can still be counted one by one.
The moment it adds up,you see how beautiful maths is.