Reasoning · Grade 5-2 Varieties of Magic Squares
Problem
Each face is bounded by four edges.
Thinking of the cube as six squares folded together — the way a net folds up — is what makes 'this edge belongs to those two faces' obvious instead of something you have to trust the drawing for.
6.G.A.4Visualize Spatial RelationshipsAdding all six double-counts each number, giving 26.
This is the whole key. Counting the same twelve numbers a second time on purpose is not a mistake — it is what lets you compare 'six equal face totals' with 'all the numbers, twice' and read the total straight off. Adding 1 to 12 and dividing 156 by 6 is Grade 4 arithmetic.
4.OA.A.3Change Focus Count The ComplementAdding all six face totals at once counts every edge exactly twice, which hands over the face total.
Each edge of a cube bounds exactly two faces, so it is counted once from each of them.
The six equal face totals put together are therefore twice all the edge numbers added up, so one face total falls out by dividing.
The top face's blank is 11.
With the total known, a face carrying three numbers is a one-subtraction subproblem. Always look for the face with the fewest blanks and do that one next.
3.NBT.A.2Identify SubproblemsThe front face's blank is 3.
Writing 11 on the top front edge is what turned the front face from a two-blank face into a one-blank face — that chain reaction is why filling the fullest face first pays off.
3.NBT.A.2Identify SubproblemsThe left face's blank is 6.
Three faces are now complete and the six numbers used so far are 1, 2, 4, 5, 8, 12 plus 11, 3, 6 — nine of the twelve. Only three circles are left.
3.NBT.A.2Identify SubproblemsThe last three faces each have two blanks.
Reducing each face to a pair total is worth doing even when it does not finish anything — three small pair totals are much easier to handle than three whole faces.
3.NBT.A.2Identify SubproblemsDouble-counting again gives 10, 9 and 7.
The same trick that found 26 in the first place works again on a smaller scale. And 7, 9 and 10 are exactly the three numbers still unused, which is a strong sign nothing has gone wrong.
4.OA.A.3Change Focus Count The ComplementAll six faces total 26.
The back and bottom faces are the ones a flat drawing hides, so turning a real cube in your hands is the safest way to be sure you added the right four circles.
6.G.A.4Create A Physical RepresentationEvery edge of a cube belongs to two faces, so adding all six faces counts each number twice — that doubling is what tells you the face total is 26.