Reasoning · Grade 5-2 Nets of a Cube

Problem

Draw cube nets inside a bounded grid

Four identical boards are printed, each a 3-column by 4-row grid with the middle cell of the bottom row cut away, leaving 11 usable cells. On each board six cells are shaded into a cube net. Nets that match after a turn or a flip count as the same shape. Draw four nets, all different.
Your answer
How to solve
Strategy Make a Systematic List — Doodling shapes on the boards until four of them work is slow and gives no way of knowing whether a fifth answer was missed. There is a much better starting point: the complete list of the 11 nets of a cube is already known from this unit, so this is a genuinely finite candidate set of 11 — exactly the situation where eliminating possibilities is honest work rather than guessing. I sort the 11 nets by their longest straight run of squares, measure the smallest rectangle each one needs, and cross off every net that cannot possibly sit inside a 3-wide, 4-tall board with its bottom middle cell missing. Whatever survives gets drawn, and cut-out paper copies confirm each placement really fits and really folds.
1STEP 1

Read the board carefully

Only 11 cells are usable.

3 × 3 + 2 = 11 usable cells, 11 - 6 = 5 left blank
2STEP 2

Sort the 11 nets by their longest straight run

Sort the 11 nets by their longest run.

6 + 3 + 1 + 1 = 11
3STEP 3

Cross off the two-rows-of-3 net — it is too long

The two-rows-of-three net is too long.

2 × 5 needed, board is 3 × 4, 5 ＞ 4
4STEP 4

Cross off all six strip-of-4 nets — the missing cell blocks them

The six strip-of-four nets are blocked by the missing cell.

strip of 4 in column 2 → needs (4,2), (4,2) ∉ board
5STEP 5

Count the survivors — exactly four, and there are exactly four boards

That leaves exactly 4, one per board.

11 - 1 - 6 = 4
6STEP 6

Draw the four nets on the four boards

Draw one on each board.

7STEP 7

Check the four are really four different shapes

The four nets are all different.

Answer
4 nets
11 − 1 − 6 = 4
Every drawn net uses exactly 6 of the 11 cells, leaves 5 blank, and avoids the cut-away cell (4,2) — checkable cell by cell. The elimination adds up: 1 net too long, 6 nets blocked by the missing cell, 4 left, and 11 - 1 - 6 = 4 matches the number of boards exactly, which is a strong sign the reasoning is complete rather than lucky. The four shapes really are different, since three of them have a longest straight run of 3 with the odd square in three different places, and the fourth has a longest run of only 2. And each one folds shut into a cube when cut out, so they are nets and not just shapes that fit.
Takeaway

Start from the 11 nets you already know, throw out the ones too big for the board or blocked by the missing cell, and exactly four survive — one for each board!

  • Read the board carefully
  • Sort the 11 nets by their longest straight run
  • Cross off the two-rows-of-3 net — it is too long
  • Cross off all six strip-of-4 nets — the missing cell blocks them
  • Count the survivors — exactly four, and there are exactly four boards
  • Draw the four nets on the four boards
  • Check the four are really four different shapes