Answers to Alex’s and Leon’s Puzzles

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. Figure them out for yourself — and then check the answers just to prove that you got them right.

Leonhard’s block puzzles

Alex’s & Leon’s homeschool puzzle

Leonhard’s Block Puzzles

Puzzle #1: Each block has 6 faces, 3 of which meet at every corner. Leon painted all his blocks the same way. He made the 3 sides that meet at one corner red and those that meet at the opposite corner forest green. When he puts the 8 blocks together, he can arrange them so that either the red or the green corners are showing.

Puzzle #2: As Alex and Leon calculated, there are 162 faces on the 27 small blocks. These are enough to cover the surface area of the larger cube 3 times. Therefore, at least in theory, there should be a way to paint these faces that would allow Alex to have her three-color puzzle cube. But how? Ah, that is puzzle #4…

Puzzle #3: Leon needed to cut a cube of wood into 27 smaller blocks to make the puzzle for his sister. If he carefully clamped the pieces back together after cutting them, the absolute minimum number of cuts he would have to make is 6, along the lines shown in the picture. In real life, it is easier to make a few extra cuts than to keep all the little pieces clamped.

Puzzle #4: Alex wanted to paint her puzzle teal, purple, and pale yellow. For each color, she will need 8 corner blocks with three faces painted, 12 edge blocks with two faces painted, and 6 blocks with one face painted. [Can you see each type of block on the cube above?) When a block has more than one face the same color, these faces must meet each other at an edge or corner.

This makes a total of 26 blocks for each color. The block in the very center of the cube will not show, so it will not have any faces painted in the exterior color. Alex and Leon will have to paint the blocks like this:

3 center blocks can be painted, each with 2 corner colors, as on Leon’s puzzle #1. The block that is painted with three teal faces and three yellow faces will go in the center of the purple puzzle cube, etc. This takes care of 2 out of the 8 corner cubes needed for each color.
Each color needs 6 more corners, for a total of 18 blocks. On each of these blocks, paint the extra sides 2 of one color and 1 of the other. For instance, on the 6 blocks with a purple corner, paint 3 of them with a teal edge (2 faces) and a single yellow face. Paint the other 3 just the opposite, with two yellow faces that meet at an edge, plus single teal face. [Other combinations will also work. Just make sure to end up with 6 single faces and 6 edges of each color.] This finishes all the corner blocks, and it also supplies all the single-face blocks needed in each color.
Finally, the last 6 blocks must each be painted as edgers in all 3 colors. Each of these will have 2 yellow sides, 2 purple sides, and 2 teal sides — and remember that the matching sides must meet at an edge!

Alex’s and Leon’s Homeschool Puzzle

The trick to solving this problem is to realize that the boy would not count himself among the brothers, nor the girl count herself as a sister. The new family has 7 children: 3 boys and 4 girls.