9.4 Paper Folding, Cube Nets, and Orthographic Views

9.4 Paper Folding, Cube Nets, and Orthographic Views

These three formats all ask the same underlying question: what does a flat thing look like once it has been assembled, or what does a solid thing look like once it has been flattened. They reward procedure rather than imagination, which is good news, because a procedure can be practised. Adamson University publishes no content outline for the AdUCET aptitude test, so treat the balance of item types as unknown and learn all three. Turning and flipping a single figure is covered in section 9.3; nothing here depends on it.

Paper folding and hole punching

The layer rule

$L = 2^{f}$

where $f$ is the number of full halving folds. One fold gives 2 layers, two folds 4, three folds 8, four folds 16. A single punch that passes through all $L$ layers and misses every crease produces exactly $L$ holes.

Both conditions in that sentence carry weight:

• Full halving folds. Folding a corner over, or folding one third of the sheet across, does not double the layers uniformly — part of the sheet stays single-layered, and a punch there yields one hole.
• Misses every crease. A punch sitting on a fold line behaves differently, as shown below.

The backward-unfolding method

Never fold forward in your head and try to imagine the punch. Work backwards.

1. Mark the punch on the final folded packet.
2. Undo the last fold first. Copy every hole you already have, mirrored across the crease you have just opened.
3. Repeat for each earlier fold in reverse order, doubling the hole count each time.
4. Stop when the sheet is flat. You should be holding $2^{f}$ holes if the punch was clear of every crease.

The reverse order is not optional. Reflections across two perpendicular halving creases happen to commute, so a careless candidate gets away with it there; reflections across two parallel creases do not commute, and unfolding in the wrong order then puts the holes in the wrong place. Build the habit on the easy cases.

Worked unfolding: one fold

Use a 4-by-4 reference grid, columns c1 to c4 from the left and rows r1 to r4 from the top. Fold the right half onto the left half, so the crease is the vertical line between c2 and c3, and punch the packet at row 2, column 2.

  Folded packet (2 layers)         Unfolded sheet (2 holes)
      c1  c2 |                       c1  c2 | c3  c4
  r1   .   . |                   r1   .   . |  .   .
  r2   .   o |  <- punch         r2   .   o |  o   .
  r3   .   . |                   r3   .   . |  .   .
  r4   .   . |                   r4   .   . |  .   .

The hole in c2 mirrors into c3, the cell on the far side of the crease.

Worked unfolding: two folds

Fold the bottom half up (crease between r2 and r3), then fold the right half onto the left (crease between c2 and c3). The packet is a 2-by-2 quarter of four layers. Punch it at r1, c1.

Undo the second fold first, mirroring c1 into c4: holes at (r1, c1) and (r1, c4). Now undo the first fold, mirroring r1 into r4: holes at (r1, c1), (r1, c4), (r4, c1) and (r4, c4).

     c1  c2  c3  c4
 r1   o   .   .   o
 r2   .   .   .   .
 r3   .   .   .   .
 r4   o   .   .   o

Four holes, one in each corner, which is $2^{2} = 4$ as required.

Worked unfolding: three folds

Carry on from that 2-by-2 packet and fold it in half once more, bottom to top, so the new crease lies between r1 and r2 of the packet. The packet is now a single row of two cells, eight layers thick. Punch at r1, c1.

Undo fold 3, mirroring r1 into r2: holes at (r1, c1) and (r2, c1). Undo fold 2, mirroring c1 into c4: add (r1, c4) and (r2, c4). Undo fold 1, mirroring r1 into r4 and r2 into r3: add (r4, c1), (r3, c1), (r4, c4) and (r3, c4).

     c1  c2  c3  c4
 r1   o   .   .   o
 r2   o   .   .   o
 r3   o   .   .   o
 r4   o   .   .   o

Eight holes, filling the whole of the first and last columns, which is $2^{3} = 8$.

Punches that sit on a crease

A punch whose centre lies on a fold line is shared by the two layers that meet at that crease, so unfolding does not double it — the hole and its mirror image are the same hole.

$\text{holes} = \frac{2^{f}}{2^{c}}$

where $c$ is the number of creases the punch straddles. Three folds with the punch on one crease gives 4 holes rather than 8; two folds with the punch on the intersection of both creases gives a single hole at the centre of the sheet. A stated hole count that is smaller than the layer count is almost always a crease-straddling item rather than an error in the question.

Cube nets

A net is a flat arrangement of six edge-joined squares that folds into a cube. There are exactly eleven distinct cube nets, counting a net and its rotations or mirror images as one. They fall into four families.

Finding opposite faces

Apply three rules, in this order:

1. Shared edge means adjacent. Two squares touching along an edge in the net are adjacent faces on the cube and can never be opposite.
2. Straight-line skip-one means opposite. In a straight run of squares, two squares with exactly one square between them fold to opposite faces.
3. L-bend skip-one means adjacent. If the path between two squares one apart turns a corner, they end up adjacent, not opposite.

Then finish by elimination: every face is adjacent to exactly four others and opposite exactly one, so once two pairs are fixed the two remaining squares must form the third pair.

        +---+
        | E |
    +---+---+---+---+
    | A | B | C | D |
    +---+---+---+---+
        | F |
        +---+

In this 1-4-1 net, rule 2 gives A opposite C and B opposite D; E and F are all that remain, so E is opposite F. The result is self-checking: E and F both touch B, so both must be adjacent to B, and indeed neither is paired with it.

The rolling method

The staircase net defeats rules 1 to 3 on their own because it contains no straight run of three. Use the rolling method: stand a cube on one square and roll it square by square, recording which of the six directions each square corresponds to. Each roll brings whichever face was pointing in the direction of travel down onto the new square. Rolls to the north or south leave the east and west faces alone, and rolls to the east or west leave the north and south faces alone, so the bookkeeping stays simple.

  +---+---+                 Direction each square turns out to be
  | P | Q |                 -----------------------------------
  +---+---+---+             R = bottom  (the starting square)
      | R | S |             Q = north   (roll north from R)
      +---+---+---+         P = west    (roll north, then west)
          | T | U |         S = east    (roll east from R)
          +---+---+         T = south   (roll east, then south)
                            U = top     (roll east, south, east)

Opposite directions now give the pairs directly: bottom with top is R opposite U, north with south is Q opposite T, and east with west is P opposite S.

Corner views and symbol orientation

• Opposite faces never share an edge and never appear together in a corner view. If a picture of the assembled cube shows three faces at once, those three are mutually adjacent, so any answer pairing two of them as opposites is wrong on sight.
• An edge shared by two squares in the net is still shared after folding. To work out which way a symbol points on the finished cube, note which edge of its square touches which neighbour and keep that relationship. An arrow pointing towards the shared edge in the net still points towards that neighbouring face on the cube.

Dice nets

Dice items add one convention: on a standard die, opposite faces sum to 7, so the pairs are 1 with 6, 2 with 5 and 3 with 4.

        +---+
        | 3 |
    +---+---+---+---+
    | 2 | 1 | 5 | 6 |
    +---+---+---+---+
        | 4 |
        +---+

Skip-one along the strip gives 2 opposite 5 and 1 opposite 6; the leftovers 3 and 4 pair up. All three sums are 7, so this net folds into a standard die. Now swap the 3 above the strip with the 5 inside it, so that the strip reads 2, 1, 3, 6 with 5 above the 1. The paper still folds into a perfectly good cube, but the pairs become 2 with 3, 1 with 6 and 5 with 4, summing to 5, 7 and 9. It is no longer a standard die, and that is exactly how the item "which of these could not be a standard die?" is built.

Orthographic views

Three standard views of a unit-cube structure: the top view looking straight down, the front view looking horizontally at the front, and the side view, usually from the left. The efficient representation to build in your head is a height map, a grid of the footprint carrying the number of cubes stacked on each cell.

  Height map (columns left to right, rows back to front)

              c1   c2   c3
     back      1    0    0
     middle    2    1    0
     front     3    2    1

• Top view is the footprint: every cell with a height of at least 1.
• Front view takes, for each column, the largest height in that column: 3, 2, 1.
• Left side view takes, for each row, the largest height in that row: back 1, middle 2, front 3. Because you are standing on the left looking right, the front row appears on the right of that view.

   Top view          Front view        Left side view
   X . .              X . .              . . X
   X X .              X X .              . X X
   X X X              X X X              X X X

Counting blocks and hidden supports

That structure holds $1+0+0+2+1+0+3+2+1 = 10$ cubes, yet the top view shows only six occupied cells. The top view can never report height, and the cubes buried under an elevated one are still there. Unless an item explicitly permits floating blocks, a cube at level 3 must be supported by cubes at levels 1 and 2 beneath it, and those hidden supports count.

Reconstructing a solid from its views

Views alone rarely fix a structure exactly; they fix a range. Suppose the top view is a full 3-by-3 square, the front view has column heights 3, 2, 1 from the left, and the left side view has heights 3, 2, 1 reading front to back.

• Maximum. Each cell can be as tall as the smaller of its column maximum and its row maximum, giving front row 3, 2, 1; middle row 2, 2, 1; back row 1, 1, 1 — a total of 14 cubes.
• Minimum. All nine footprint cells need at least one cube. Column 1 must reach 3 somewhere and the front row must reach 3 somewhere, and the front-left corner cell does both jobs at once for 2 extra cubes; likewise one cell can serve column 2 and the middle row at height 2 for 1 extra. Total 12 cubes.

So "how many blocks are in this structure?" is answerable only when the item supplies a picture as well as views, or asks explicitly for the minimum.

Common traps

• Assuming every fold halves the sheet. Corner folds and diagonal folds do not.
• Unfolding in the original order instead of reverse order.
• Forgetting that a punch on a crease halves the hole count.
• Treating two squares that share an edge in a net as opposite faces. They never are.
• Reading a corner view of a cube and pairing two visible faces as opposites.
• Counting the cells of the top view and calling that the block total.
• Forgetting the hidden support cubes underneath an elevated block.