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Problem of the Week
Problem C
Counting Numbered Cubes

Dmitri has a collection of identical cubes. Each cube is labelled with the integers \(1\) to \(6\), as shown in the following net:

Net has four squares arranged in a horizontal row, labelled 4, 6, 3, 1 from left to right. The square labelled 6 has a square labelled 5 above it and a square labelled 2 below it.

(This net can be folded to make a cube.)

He forms a pyramid by stacking layers of the cubes on a table, as shown, with the bottom layer being a \(7\) by \(7\) square of cubes.

The pyramid has four layers. Each layer above the bottom layer, which is a 7 by 7 square of cubes, is centred on top of the layer below it. The other layers, from bottom to top, are a 5 by 5 square of cubes, then a 3 by 3 square of cubes, then 1 cube.

  1. Determine the total number of cubes used to build the pyramid.

  2. How many faces are visible after the pyramid is built and sitting on the table?

  3. He wants to position the cubes so that when all of the visible numbers are added up, the total is as large as possible. What is this total?

Themes: Geometry & Measurement, Number Sense