Here are clues on how to solve the New Hampshire State Math Contest puzzle box. It is suggested that you open the hints one at a time as needed. The goal is to fit all of the blocks into the original box.
Clue #1:
Take an inventory of all the blocks – how many types, how many of each type.
Clue #2:
Call the smallest block a 1x1x1. What are the dimensions of all the other blocks in terms of the 1x1x1 block?
Clue #3:
What are the overall dimensions of the box interior? What is the volume of the box interior? What is the sum of the volumes of all the individual blocks?
Clue #4:
Define a slice as a 5x5x1 cross section of the box interior. How many slices are there? What is the volume of each slice?
Clue #5:
All 15 slices (top to bottom, left to right, front to back) must be filled with filled with the cross sections of various blocks.
Clue #6:
Each slice has a volume of 25.
Clue #7:
The 1x2x4 blocks have cross sections with volume 2, 4, or 8. The 2x2x3 blocks have cross sections of volume 4 or 6. The 1x1x1 blocks have cross section of volume 1.
Clue #8:
The box slices have a volume of 25 — and 25 is an odd number.
Clue #9:
Since each slice has a volume of 25 (an odd number), and cross section volumes of the larger blocks are all even numbers, there must be a 1x1x1 block in every slice.
Clue #10:
One way to ensure that one (and only one) 1x1x1 cube is in every slice is to arrange them in a stepped diagonal that goes from the front left bottom corner to the back right top corner, opposite of it in the box interior. The larger blocks then have to fit around this placement of the 1x1x1 cubes.
Complete Solution:
Place two 1x2x4 blocks, one 2x2x3 block, three 1x1x1 blocks, as shown:
Add two 1x1x1, one 1x2x4, three 2x2x3 blocks on top:
Add three 1x2x4, two 2x2x3 blocks on top:
This puzzle is a slightly modified version of Conway’s Packing Puzzle, invented by legendary Cambridge mathematician John Horton Conway(1937-2020).
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