TACHS: Matrices: combine dots by overlay
Learn to read a six-circle figure as a list of dark spots, confirm the overlay rule from a shared spot, and build the missing figure one position at a time.
How do you build the third figure when it combines the dark circles of the first two?
Short answer
Stack the first two figures: a circle is dark in the third figure if it is dark in either one. Confirm the rule with a spot that is dark in both, then check every position, not the count.
Know first: Left and right, Reading a small figure in a fixed order
Step 1 of 5
The third figure is the first two stacked
In an overlay matrix, each cell shows six small circles in two columns of three. Name the positions upper-left, upper-right, middle-left, middle-right, lower-left and lower-right. Each circle is dark or white.
The third figure in a row is what you would see if you stacked the first two figures on a window: a circle is dark if it is dark in either figure. The same rule usually works down the columns too.
| Circle in figure 1 | Circle in figure 2 | Circle in figure 3 |
|---|---|---|
| dark | white | dark |
| white | dark | dark |
| dark | dark | dark |
| white | white | white |
The third row is the one that matters. A shared dark circle stays dark under overlay.
Step 2 of 5
Build the missing figure spot by spot
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | upper-left | lower-right | upper-left, lower-right |
| Row 2 | middle-right | middle-right, lower-left | middle-right, lower-left |
| Row 3 | upper-left, middle-right | middle-right, lower-left, lower-right | ? |
Each entry lists the dark circles. Every other circle in that cell is white.
Worked example: Overlay a bottom row
Use the matrix above. Which figure belongs in the empty bottom-right cell?
- Aupper-left, lower-left, lower-right
- Bupper-right, middle-right, lower-left, lower-right
- Cupper-left, middle-right, lower-left, lower-right
- Dmiddle-right, lower-left, lower-right
- Eall six dark
- 1
Test the top row
Upper-left, then lower-right, then both. The first two figures share no spot, so both overlay and cancel explain this row. Keep looking.
- 2
Test a row with a shared spot
Middle row: middle-right, then middle-right and lower-left, then middle-right and lower-left. Middle-right was dark in both and stayed dark. The rule is overlay.
- 3
Stack the bottom row
Figure 1: upper-left, middle-right. Figure 2: middle-right, lower-left, lower-right. Dark in either: upper-left, middle-right, lower-left, lower-right.
- 4
Check down the right column
Upper-left and lower-right, stacked with middle-right and lower-left, gives the same four spots.
- 5
See why the other choices are there
Upper-left, lower-left, lower-right uses the cancel rule and drops the shared middle-right. Upper-right, middle-right, lower-left, lower-right has four dark circles but swaps upper-left for upper-right. Middle-right, lower-left, lower-right copies the cell to the left. All six dark adds circles that neither figure had.
Answer
Dark circles at upper-left, middle-right, lower-left and lower-right.
Find the wrong step
A student checks a choice for a row whose first two figures are upper-right, middle-left and middle-left, lower-right. The choice shows three dark circles: upper-right, middle-left and lower-left.
- 1
Stacks the two figures and gets upper-right, middle-left and lower-right.
- 2
Sees that the choice also has three dark circles and accepts it.
What went wrong
Matching the count is not matching the figure. The choice has lower-left where the stack needs lower-right.
- 3
Moves on to the next question.
The fix
Compare position by position. The choice fails at lower-left versus lower-right, so it is out.
Step 3 of 5
Overlay on your own
Check yourself · Question 1
In one row of an overlay matrix, figure 1 has dark circles at upper-right and middle-left. Figure 2 has dark circles at upper-right and lower-right. Which figure comes third?
Answer: C
A circle is dark if it is dark in either figure: upper-right (both), middle-left (figure 1) and lower-right (figure 2).
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | upper-left | upper-left, middle-right | upper-left, middle-right |
| Row 2 | lower-right | middle-right | middle-right, lower-right |
| Row 3 | upper-left, lower-right | upper-left, middle-right | ? |
Each entry lists the dark circles. Every other circle in that cell is white.
Check yourself · Question 2
Use the matrix above. Which figure belongs in the empty bottom-right cell?
Answer: D
The top row has a shared upper-left circle that stays dark, so the rule is overlay. Bottom row: upper-left and lower-right, stacked with upper-left and middle-right, gives upper-left, middle-right and lower-right. The right column gives the same three.
Trap answer: middle-right, lower-right
Wrong rule for a shared spot
- Why it looks right
- Two dark circles look tidy, and if you skip the rule check, the cancel rule seems just as likely as overlay.
- Why it's wrong
- In the top row, upper-left is dark in both figures and stays dark in the third. That makes the rule overlay, so shared circles stay.
- The right answer
- upper-left, middle-right, lower-right.
Common mistake
Two choices both have three dark circles, so I just picked the one that looked closest.
- Why it's tempting
- Small figures look alike, and counting is faster than checking six spots.
- Do this instead
- Read each figure in the same order every time: upper row left to right, then middle, then lower. A choice is right only if every position matches.
Step 4 of 5
What to remember
Remember
- Overlay: a circle is dark in the third figure if it is dark in either of the first two.
- Use a shared dark spot in a completed row to tell overlay from the cancel rule.
- Compare position by position. The right number of dark circles is not enough.
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, which spot was dark in both of the first two figures of a row, and what happened to it in the third?