DILR
Games & Tournaments
A one-page CAT 2026 reference for games and tournaments, covering knockout and round-robin match counts, byes, double round-robins, points table reconciliation and scoring, qualification ceilings,…
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Cubes
CAT’26 DILR CHEATSHEET
Every painted cube formula and shortcut you need for CAT 2026.
Painted cube questions look like a geometry topic and solve like pure counting, and that mismatch is exactly where most students slow down. A big cube gets painted on the outside, sliced into a grid of n smaller cubes per edge, and the question wants to know how many small cubes ended up with 3, 2, 1 or 0 painted faces. Once you know that corners are always 8, edges scale as 12(n minus 2), faces as 6(n minus 2) squared and the interior as (n minus 2) cubed, most of these questions turn into direct substitution. The harder versions swap in a cuboid, paint only some faces, or replace the grid cut with straight slicing, each needing its own small adjustment. This sheet walks through every version, with one worked example per box so you drill the setup and not just the numbers.
Every cubes rule you need
1The Basic Setup
A painted cube is cut into n equal smaller cubes along each edge.
Total small cubes = n3
Example. A cube painted on all faces and cut into 4 equal pieces per edge gives 43 = 64 small cubes.
CAT Hack. n is the number of cuts per edge plus one, so a cube cut twice on each edge gives n = 3, not n = 2.
2Corner Cubes (3 Faces Painted)
Small cubes sitting at a corner of the big cube show paint on 3 sides.
3 faces painted = 8, always, for any n ≥ 2
Example. Whether the big cube is cut into 27 or 1000 pieces, exactly 8 small cubes sit at the 8 corners with 3 painted faces.
CAT Favourite. The corner count never changes with n. Every cube, however finely cut, has exactly 8 corners.
3Edge Cubes (2 Faces Painted)
Small cubes along an edge, excluding corners, show paint on 2 sides.
2 faces painted = 12(n − 2)
Example. For n = 5, the 2-face count is 12(5−2) = 12 × 3 = 36 small cubes.
CAT Hack. 12 comes from the cube's 12 edges, and each edge contributes (n−2) such cubes once the 2 corners are removed.
4Face Cubes (1 Face Painted)
Small cubes in the middle of a face, not touching any edge, show 1 painted side.
1 face painted = 6(n − 2)2
Example. For n = 5, the 1-face count is 6(5−2)2 = 6 × 9 = 54 small cubes.
CAT Hack. 6 comes from the cube's 6 faces, and each face contributes an (n−2) by (n−2) inner square of such cubes.
5Interior Cubes (0 Faces Painted)
Small cubes fully inside the big cube, touching no outer face at all.
0 faces painted = (n − 2)3
Example. For n = 5, the unpainted core is (5−2)3 = 33 = 27 small cubes.
CAT Favourite. This inner (n−2)3 block is itself a smaller cube sitting exactly in the centre, one layer removed on every side.
6Reconciliation Check
The four painted-face categories must add back up to the total cube count.
8 + 12(n−2) + 6(n−2)2 + (n−2)3 = n3
Example. For n = 5, that is 8 + 36 + 54 + 27 = 125, which matches 53 exactly.
CAT Hack. After computing all four categories, add them up and check the total equals n cubed. It catches almost every arithmetic slip.
7Cube Painted on Fewer Than 6 Faces
Some questions paint only some faces before cutting, changing every count.
Redo each category by checking which painted faces touch that cube
Example. If only the top face is painted, no small cube can show 3 painted sides at all, since a corner needs 3 mutually painted faces meeting it, so that category drops to zero.
Common Mistake. Never reuse the standard 8, 12(n−2), 6(n−2)² formulas when fewer than 6 faces are painted. Rebuild the count from the actual painted faces.
8Two Opposite Faces Painted
Only a pair of opposite faces, like top and bottom, are painted before cutting.
1 face painted = 2n2, 0 faces painted = n2(n−2)
Example. For n = 4 with only top and bottom painted, the 1-face count is 2(4)2 = 32 small cubes.
CAT Insight. With only opposite faces painted, no small cube can ever show 2 or 3 painted faces, since painted faces never meet at an edge or corner.
9Two Adjacent Faces Painted
Two faces that share an edge, like top and front, are painted before cutting.
2 faces painted = (n − 2), the cubes along the shared edge
Example. For n = 5 with two adjacent faces painted, exactly 5−2 = 3 small cubes sit on the shared edge with 2 painted sides.
CAT Insight. Adjacent painted faces create exactly one shared edge of double-painted cubes, unlike opposite faces which create none.
10Cuboid Instead of a Cube
A painted rectangular box is cut into unit-sized pieces along unequal edges.
Small pieces = l × b × h, using the piece counts per edge
Example. A box cut into a 4 by 3 by 2 grid of pieces has 4 × 3 × 2 = 24 small pieces total.
Common Mistake. A cuboid needs 3 separate edge counts, l, b and h. Do not force the single-variable cube formulas onto it.
11Cuboid Face and Corner Counts
The 8, 12 and 6 categories still apply to a cuboid, but each term uses different edges.
3 faces = 8, 2 faces = 4(l−2)+4(b−2)+4(h−2), 0 faces = (l−2)(b−2)(h−2)
Example. For a 6 by 4 by 3 cuboid, the fully interior count is (6−2)(4−2)(3−2) = 4 × 2 × 1 = 8 pieces.
CAT Hack. The 8 corners stay fixed just like a cube, but every other category needs each of the three edge lengths plugged in separately.
12Cubes with No Paint on a Given Face
Counting small cubes that avoid paint on one specific chosen face.
Avoids one face = n3 minus the cubes touching that face
Example. A small cube layer touching one face of an n by n by n cube has n2 cubes, so avoiding that face leaves n3 − n2 = n2(n−1) cubes.
CAT Insight. This is really just removing one outer layer of the cube, a single n by n sheet, from the total.
13Slicing Instead of a Grid Cut
Straight parallel cuts through a solid, not a full grid, create fewer pieces.
k straight cuts, all parallel, create k + 1 pieces
Example. Making 5 parallel cuts through a cube produces 5 + 1 = 6 slabs, regardless of the cube's actual size.
Common Mistake. Parallel slicing is a completely different setup from the grid-cut cube. Do not apply n³ style formulas to a slicing question.
Exam shortcuts, common traps and revision
14
CAT Exam Shortcuts
- 3 faces painted = 8, fixed for any n
- 2 faces painted = 12(n−2)
- 1 face painted = 6(n−2)2
- 0 faces painted = (n−2)3
- All four categories always sum to n3
- Fewer than 6 faces painted needs a fresh count, not the standard formulas
- A cuboid needs l, b and h handled separately, never a single n
15
Most Common CAT Traps
- Confusing n, the pieces per edge, with the number of cuts made
- Reusing the 6-face formulas when only some faces are painted
- Treating opposite painted faces as if they could create 2 or 3 painted cubes
- Forcing a cuboid into the single-variable cube formulas
- Skipping the reconciliation check and missing an arithmetic slip
- Applying grid-cut cube formulas to a straight parallel-slicing question
16
30-Second Revision Box
- 8 corners, 12(n−2) edges, 6(n−2)2 faces, (n−2)3 core
- These four always add up to n3
- Corner count of 8 never changes with n
- Opposite faces painted never create 2 or 3 painted cubes
- Cuboids need l, b, h plugged in separately
- Parallel slicing gives cuts + 1 pieces, a different rule entirely
The single habit that prevents most errors here is the reconciliation check, adding all four painted-face categories back up and confirming they equal n cubed, since a mismatch catches an arithmetic slip before it costs you the question. Read carefully whether all 6 faces are painted, whether the solid is a cube or a cuboid, and whether it is a grid cut or a straight slice. Run a full mock and check your projected percentile with the Optima Learn CAT score predictor, then work through the DILR breakdowns on the blog. If you would rather build a plan with a person, book a free call with an Optima Learn mentor.
