You are wandering around in the fourth dimension (see "Hyperspace Hangout"), going in and out of weird holes in the walls, when suddenly a phone rings.
Puzzled, you find the receiver and answer, "Hello."
"This is Digit Three, your old friend from Pi Land (see "Pi in the Sky")."
"Fantastic! Can you help us?" you ask. "We seem to be stuck in the fourth dimension, and we have no idea how to get back into regular space. Will we ever see the stars again? Will we ever return to Earth?"
"That crazy Eight is up to some nasty tricks again," says Three. "When I heard that Eight had sent you guys to the Cubic Grid Galaxy (see "Galactic Gridlock"), I knew that you would have a knotty problem and would need my help to get you out."
"How do we get out of here? We need to be back in three-dimensional space!"
"Check each corner of your cubic space capsule. In one of the corners you will find a pile of thirty-six cubes about the size of children's building blocks. Arrange the blocks into a perfect triangle. Good luck!"
You hear a click and realize that Three has hung up, so you, Bill, and Anita go and find the thirty-six blocks. How can you arrange them in a perfect triangle?
TRY IT!
Arrange thirty-six cubes (or square tiles) into a triangular pattern.
The steps below show how to arrange the blocks (or tiles) into a horizontal triangle lying flat on a table or on the floor. If you are using cubes, you may stack them to form a vertical triangle, if you wish.
You will need:
- Thirty-six cubic blocks (even sugar cubes would work), or thirty-six square tiles (about 1 inch or 2.5 centimeters wide) cut from a sheet of paper or cardboard
- flat surface
What to do:
- Lay one of the cubes or squares on your surface and place two more below it so that the top cube sits on the border between the two lower cubes to form a triangle.
- Below the row of two, place a row of three cubes so your cubes form a larger triangle. Note that each cube goes beneath a crack between two cubes in the previous row, and the cubes at each end stick out.
- Make a fourth row using four cubes, and continue adding rows of cubes until you have used up all thirty-six cubes.
Six square tiles arranged to form a triangle.
Triangular Numbers
Your thirty-six-cube triangle should have eight rows. The first row has just one cube. Each row that follows has one cube more than the previous row.
If you count the total number of cubes from the top to a certain row below, you get what is known as a triangular number. The first triangular number is 1, followed by 3, 6, 10, 15, 21, 28, 36, and so on.
Counting up the total number of squares required to make up a triangular pattern of a certain number of rows gives a triangular number: 1, 3, 6, 10, 15, 21, 28, and 36.
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