TACHS: Matrices: one of each, three ways
Learn to split each cell into shape, shading and dot count, solve each feature on its own, and reject choices that get two of three right.
How do you solve a matrix where shape, shading and dot count all change at once?
Short answer
Treat it as three small puzzles. Each row and column uses every shape, every shading and every dot count once, so find the missing value of each feature, then pick the choice that matches all three.
Know first: Names of common shapes, Telling dark, light and unfilled shading apart
Step 1 of 5
Split each cell into three features
In these matrices, each cell holds one shape with a shading, plus a small row of dots under it. There are three shapes, three shadings (dark, light, unfilled) and three dot counts (1, 2, 3).
Every row and every column uses each shape once, each shading once and each dot count once. It works like three tiny Sudoku puzzles laid on top of each other. Solve them one at a time.
| Feature | Cell 1 | Cell 2 | Missing |
|---|---|---|---|
| Shape | triangle | square | hexagon |
| Shading | dark | unfilled | light |
| Dots | 3 | 1 | 2 |
Each missing value is the one the row has not used yet.
Step 2 of 5
Three small puzzles, one answer
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | dark square, 3 dots | unfilled hexagon, 1 dot | light triangle, 2 dots |
| Row 2 | light hexagon, 2 dots | dark triangle, 3 dots | unfilled square, 1 dot |
| Row 3 | unfilled triangle, 1 dot | light square, 2 dots | ? |
Dark is solid, light is a pale fill, and unfilled is an outline only. The dots sit in a row under the shape.
Worked example: Find the missing cell feature by feature
Use the matrix above. Which figure belongs in the empty bottom-right cell?
- Adark hexagon, 1 dot
- Bunfilled hexagon, 3 dots
- Cdark triangle, 3 dots
- Dlight square, 2 dots
- Edark hexagon, 3 dots
- 1
Shape
The bottom row has a triangle and a square. The missing shape is the hexagon.
- 2
Shading
The bottom row has unfilled and light. The missing shading is dark.
- 3
Dot count
The bottom row has 1 and 2. The missing count is 3.
- 4
Check the right column
Light triangle with 2 dots, unfilled square with 1 dot. Missing: hexagon, dark, 3. Same description.
- 5
See why the other choices are there
Dark hexagon, 1 dot has the wrong count. Unfilled hexagon, 3 dots has the wrong shading. Dark triangle, 3 dots has the wrong shape. Each gets two of three. Light square, 2 dots copies the cell next to the gap.
Answer
Dark hexagon with 3 dots.
Find the wrong step
A student solves a bottom row that shows a light hexagon with 1 dot, then a dark square with 3 dots, then the gap.
- 1
Shape: hexagon and square are used, so the missing shape is the triangle.
- 2
Shading: picks dark, because the triangle in the row above is dark.
What went wrong
The missing shading comes from this row. Light and dark are already used here, so the missing shading is unfilled.
- 3
Dots: 1 and 3 are used, so the missing count is 2.
The fix
Every feature comes from what the row has not used: an unfilled triangle with 2 dots.
Step 3 of 5
Check all three features
Check yourself · Question 1
A row shows a dark triangle with 2 dots, then an unfilled hexagon with 3 dots. Each row uses every shape, shading and dot count once. What belongs in the third cell?
Answer: C
Shape: triangle and hexagon are used, so square. Shading: dark and unfilled are used, so light. Dots: 2 and 3 are used, so 1.
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | light triangle, 2 dots | dark square, 3 dots | unfilled hexagon, 1 dot |
| Row 2 | unfilled square, 1 dot | light hexagon, 2 dots | dark triangle, 3 dots |
| Row 3 | dark hexagon, 3 dots | unfilled triangle, 1 dot | ? |
Dark is solid, light is a pale fill, and unfilled is an outline only. The dots sit in a row under the shape.
Check yourself · Question 2
Use the matrix above. Which figure belongs in the empty bottom-right cell?
Answer: E
Bottom row: hexagon and triangle are used, so square. Dark and unfilled are used, so light. 3 and 1 are used, so 2 dots. The right column (unfilled hexagon with 1 dot, dark triangle with 3 dots) gives the same answer.
Trap answer: light square, 3 dots
Two out of three
- Why it looks right
- The shape and shading jump out first, and both are correct.
- Why it's wrong
- The dark hexagon in the same row already has 3 dots, so 3 cannot appear again.
- The right answer
- light square, 2 dots, which matches on all three features.
Common mistake
I matched the shape and moved on.
- Why it's tempting
- Shape is the biggest, most visible feature, so it feels like the main one.
- Do this instead
- Wrong choices are built to match two features. Count the dots and name the shading for every choice you keep.
Step 4 of 5
What to remember
Remember
- Split each cell into shape, shading and dot count, and solve each feature separately.
- The missing value of each feature is the one the row, or the column, has not used.
- Check every choice on all three features. Choices that get two of three right are the usual traps.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice matrix, what did you find for shape, shading and dot count, and which choice got two of the three right?