TACHS: Matrices: dots that cancel
Learn the cancel rule, where a circle dark in both of the first two figures turns white, and build the missing figure one position at a time.
How do you find the third figure when matching dark circles erase each other?
Short answer
A circle is dark in the third figure only when it is dark in exactly one of the first two. Dark in both, or in neither, means white. Confirm the rule with a shared spot, then check every position.
Know first: Left and right, Reading a small figure in a fixed order
Step 1 of 5
Matching dark circles erase each other
These matrices use the same six-circle frame as overlay matrices: upper-left, upper-right, middle-left, middle-right, lower-left and lower-right, each dark or white.
Picture two clear sheets laid on top of each other, with one twist: wherever two dark circles line up, they erase each other. What is left is the third figure. A circle is dark only when it is dark in exactly one of the first two figures.
| Circle in figure 1 | Circle in figure 2 | Circle in figure 3 |
|---|---|---|
| dark | white | dark |
| white | dark | dark |
| dark | dark | white |
| white | white | white |
The third row is the difference from overlay. A shared dark circle disappears.
Step 2 of 5
Cancel spot by spot
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | upper-left, middle-right | upper-left, lower-left | middle-right, lower-left |
| Row 2 | upper-right | upper-right, middle-right | middle-right |
| Row 3 | upper-left, upper-right, middle-right | upper-left, upper-right, middle-right, lower-left | ? |
Each entry lists the dark circles. Every other circle in that cell is white.
Worked example: A bottom row that nearly empties
Use the matrix above. Which figure belongs in the empty bottom-right cell?
- Aupper-left, upper-right, middle-right, lower-left
- Bno dark circles
- Clower-left
- Dlower-right
- Emiddle-right, lower-left
- 1
Confirm the rule
Top row: upper-left is dark in both of the first two figures and white in the third. The rule is cancel. The middle row agrees: upper-right is shared and disappears.
- 2
Go spot by spot in the bottom row
Figure 1: upper-left, upper-right, middle-right. Figure 2: upper-left, upper-right, middle-right, lower-left.
Upper-left: in both, so it cancels.
Upper-right: in both, so it cancels.
Middle-right: in both, so it cancels.
Lower-left: only in figure 2, so it stays. - 3
Check down the right column
Middle-right and lower-left, then middle-right. Middle-right cancels and lower-left stays. Same answer.
- 4
See why the other choices are there
Upper-left, upper-right, middle-right, lower-left stacks without canceling. No dark circles cancels lower-left too, though it was in only one figure. Lower-right has one dark circle in the wrong place. Middle-right, lower-left copies the top-right cell.
Answer
One dark circle, at lower-left.
Wrong: A nearly empty choice looks unfinished, so the fuller choice must be right.
Right: A nearly empty choice is often right under the cancel rule.
Every shared circle disappears, so the third figure can have just one or two dark circles. Trust the spot-by-spot check over how full a figure looks.
Step 3 of 5
Cancel on your own
Check yourself · Question 1
Under the cancel rule, figure 1 has dark circles at upper-left, middle-left and lower-right. Figure 2 has dark circles at upper-left and middle-right. Which figure comes third?
Answer: B
Upper-left is in both, so it cancels. Middle-left and lower-right are only in figure 1, and middle-right is only in figure 2, so those three stay dark.
| Column 1 | Column 2 | Column 3 | |
|---|---|---|---|
| Row 1 | upper-right, middle-left | middle-left, lower-right | upper-right, lower-right |
| Row 2 | upper-left, upper-right | upper-right, lower-left | upper-left, lower-left |
| Row 3 | upper-left, middle-left | upper-right, middle-left, lower-left, lower-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: B
In the top row, middle-left is shared and disappears, so the rule is cancel. Bottom row: upper-left and middle-left, then upper-right, middle-left, lower-left and lower-right. Middle-left cancels and the rest stay: the four corner circles. The right column gives the same four.
Trap answer: upper-left, upper-right, middle-left, lower-left, lower-right
Stacked without canceling
- Why it looks right
- It looks complete, and every circle in it really is dark somewhere in the bottom row.
- Why it's wrong
- Middle-left is dark in both figures of the bottom row, so it erases itself.
- The right answer
- upper-left, upper-right, lower-left, lower-right, the four corners.
Common mistake
I canceled a circle because it appeared somewhere else in the row.
- Why it's tempting
- Once you start erasing, it is easy to erase too much.
- Do this instead
- Only a circle dark in both of the first two figures, at the same position, cancels. Check each of the six positions on its own.
Step 4 of 5
What to remember
Remember
- Cancel rule: a circle is dark in the third figure only if it is dark in exactly one of the first two.
- A shared dark circle in a completed row tells you whether the rule is cancel or overlay.
- Compare position by position, and expect sparse answers.
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 positions canceled in the bottom row, and which choice would you pick if you stacked without canceling?