Dices

Module 12 — Lesson 0012 · In Depth · 45 min

Dice is spatial logic: standard dice sum to 7 on opposites; ordinary dice you deduce from views. Learn the three closed-die rules and the two open-net rules — then cutting/painting is just (n−2) counting.

1. The Foundation: Types of Dice

Type	Rule	Opposites
Standard / Unbiased die	Sum of opposite faces = 7	1↔6, 2↔5, 3↔4
Ordinary (non-standard) die	Random placement — no 7 assumption	Must deduce from views

If question says "standard" or "unbiased" — use 7 immediately. If not, use visual rules below.

Standard die: the three opposite pairs are forced to sum to 7.

2. Rules for Closed Dice (Multiple Views)

Rule 1: Adjacency Rule

If you see two faces at the same time as a third face in a single view, those two faces are adjacent to the third — not opposite. By collecting neighbors from 2–3 views, you eliminate four neighbors for a given number, leaving only one possible opposite.

Rule 2: Two-Face Common Rule

If two views share two identical faces, the remaining third faces are opposite each other.

Worked: View1 shows (2,3,5), View2 shows (2,3,6) → common are 2 and 3 → 5 opposite 6.

Rule 3: One-Face Common (Clockwise / Anti-clockwise) Rule

When two views share one face, use rotation:

Identify the common face in both views.
Move clockwise (or anti-clockwise) from that face in both diagrams.
Numbers encountered at the same step are opposite each other.

One face common → clockwise pairing gives two opposite pairs at once.

3. Mastering Open Dice (Unfolded Nets)

The Alternate Face Rule

In any straight line of four boxes in a net, alternate faces (skipping one) are opposite.

Example: vertical strip 1-2-3-4 → 1 ↔ 3 and 2 ↔ 4.

The Z-Logic

For non-linear shapes, follow a Z-shaped path: start at a face, move two steps in one direction and one step at a right angle — the faces at the ends of the Z are opposite.

Straight-4 → alternate; non-linear → follow a Z.

4. Advanced Dice Properties

Vertices and Corners

Cube has 8 vertices — 3 faces meet at each corner.

Those 3 faces are never opposite each other.
On a standard die, max corner sum = 4+5+6=15, min = 1+2+3=6.

Rotational Logic

Three axes: vertical, horizontal, lateral. E.g., rotate 90° clockwise about vertical → Front→Left, Right→Front. Keep one face fixed to track.

Painting and Cutting — Cubes from a Cube

Paint a cube, cut into n³ small cubes where n = divisions per edge (e.g., 3×3×3=27):

3 faces painted: 8 corners — always 8.
2 faces painted: edge centers — 12×(n−2)
1 face painted: face centers — 6×(n−2)²
0 faces painted: inner core — (n−2)³

Same as Module 06 Cubes — now in dice context.

5. Summary of Variables

Symbol	Meaning
`S`	Sum of opposite faces (7 for standard)
`n`	Divisions per edge (painted cubes)
`V / E / F`	Vertices 8 / Edges 12 / Faces 6

Standard die opposite sum?

Two views share 2 faces (2,3) and third faces 5 vs 6 → opposite?

One face common → method?

Straight line of 4 in net: 1-2-3-4 → opposite pairs?

Z-logic is for?

Max corner sum on standard die?

3×3×3 painted cube → 1 face painted?

Notes

Dice Types :-

Standard ⇒ opposite sum 7 (1↔6,2↔5,3↔4) ; Ordinary ⇒ deduce from views

Closed Dice :-

a) Adjacency ⇒ seen together ≠ opposite

b) Two-face common ⇒ remaining are opposite (2,3 common→5↔6)

c) One-face common ⇒ clockwise from common → same steps are opposite

Open Dice :-

Straight 4 ⇒ alternate (1↔3,2↔4) ; Z ⇒ non-linear, ends of Z are opposite

Advanced :-

Vertex: 3 faces meet, max 4+5+6=15, min 1+2+3=6 ; Painting: n³ cubes: 3F→8, 2F→12(n−2), 1F→6(n−2)², 0F→(n−2)³

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