Spatial Puzzles


Strange Miles

You are somewhere on Earth. You walk due south 1 mile, then due east 1 mile, then due north 1 mile. When you finish this 3-mile walk, you are back exactly where you started.

It turns out there are an infinite number of different points on earth where you might be. Can you describe them all?

It's important to note that this set of points should contain both an infinite number of different latitudes, and an infinite number of different longitudes (though the same latitudes and longitudes can be repeated multiple times); if it doesn't, you haven't thought of all the points.


The Circular Lake

A swan sits at the center of a perfectly circular lake. At an edge of the lake stands a ravenous monster waiting to devour the swan. The monster can not enter the water, but it will run around the circumference of the lake to try to catch the swan as soon as it reaches the shore. The monster moves at 4 times the speed of the swan, and it will always move in the direction along the shore that brings it closer to the swan the quickest. Both the swan and the the monster can change directions in an instant.

The swan knows that if it can reach the lake's shore without the monster right on top of it, it can instantly escape into the surrounding forest.

How can the swan succesfully escape?


Tiling Without Corners

You can easily "tile" an 8x8 chessboard with 32 2x1 tiles, meaning that you can place these 32 tiles on the board and cover every square.

But if you take away two opposite corners from the chessboard, it becomes impossible to tile this new 62-square board.

Can you explain why tiling this board isn't possible?


Dividing the Cake

Two twin brothers share the same birthday. Their father gets them a perfectly rectangular birthday cake, and the brothers decide to split the cake into two equal halves so that they each get to eat the same amount of cake.

However, before they can divide it, their father cuts out a perfectly circular (or more precisely, cylindrical) piece from the cake and eats it.

How can the brothers divide the rest of the cake with exactly one straight-line slice? The slice must be a vertical slice straight down through the cake, and is allowed to pass through the removed circle if needed. The brothers have a ruler and a compass to help them choose where to slice the cake.


9 Dots, 4 Lines

Look at the 9 dots in this image. Can you draw 4 straight lines, without picking up your pen, that go through all 9 dots?