Leonhard Jones is Alexandria Jones’s younger brother. He enjoys woodworking, and he cut a wooden cube into 8 smaller blocks to make himself a puzzle.
Leon painted the 8 blocks with his two favorite colors: red and forest green. When he was finished, Leon could put the blocks together into a red cube, or he could switch them around to make a green cube.
How did Leon paint his blocks?
Alex liked Leon’s new block puzzle so much that she wanted him to make a similar puzzle for her.
“The problem in, I have three favorite colors,” she said. “Can you make a set of blocks that will turn into a teal cube, or a purple cube, or one that’s pale yellow?”
Leon frowned. “That’s tricky. I know I can’t do it like I did mine.”
“What if you tried a cube, made from 27 small blocks?”
“Oh, right! If I can color a puzzle with two colors, then a puzzle should be able to take three colors. Probably. That is… maybe. I’m not sure.”
What do you think? Can you prove that it will work? (Answer before you read the rest of the story!)
Puzzle #2, Partially Answered
Alex nibbled on her lip and stared at the ceiling — her thinking expression.
“Hmm…,” she said. “Each block has 6 square faces, which means you would have squares to paint.”
“Hey, Alex. You would help with the painting, wouldn’t you?”
“If you want.”
“Okay, then,” said Leon. “We need enough squares of each color to cover the surface of the cube. That’s 9 squares per side times 6 sides — 54 squares. So we have to paint 54 squares teal, 54 purple, and 54 yellow.
“And also,” Alex said. “It should work.”
Leon nodded. “So let’s do it. I have an extra block of wood out in the garage.”
But that’s only part of the answer. Can you figure out the rest? See puzzle #4, below…
Alex followed her brother out to the tool bench, with her trusty dog Ramus bounding along behind. But when the children started poking around the woodpile, Rammy decided to run out to the backyard and chase a few squirrels.
Leon laughed. “He hates the saw.”
“Yeah. I think it hurts his ears.”
They found the block of wood. It was the end of an old 4×4, and it made a nearly perfect cube.
“Here’s a question for you,” Leon said. “I have to cut this big cube into 27 little blocks for your puzzle. But pretend I’m so lazy that I want to do it the shortest way possible. What is the least number of cuts I need to make?”
“Well, it’s obviously not 27, or you wouldn’t be asking, right?”
“Right. It’s way less than 27.”
Do you know the answer to Leon’s question?
They spent the rest of the afternoon measuring, cutting, and sanding the blocks. Alex biked in to the paint store and bought three small cans of latex paint. After dinner, they spread newspaper on the table, but then they had to leave for a homeschool group meeting.
Finally, the next morning, they got their brushes out and filled an old butter dish with water. Alex and Leon were ready to paint.
Then Leon hid his face in his hands and groaned.
“What’s wrong?” Alex asked.
“I just thought of something. Which squares do we paint with which colors?”
Is it possible to paint 27 small cubes so that they can be arranged into a larger cube 3 different ways, each time with a different color covering the outside?
Edited to Add
The answers to these puzzles are now posted:
To Be Continued…
Read all the posts from the January/February 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.