TACHS: Classification: compare inside and outside counts
Learn to count the dots inside and outside each outline, test simple rules until one fits all three examples, and ignore the outline's shape.
How do you find the rule when the examples have different outlines and different numbers of dots?
Short answer
Write each example as a pair, inside count and outside count. Test a difference first, then a multiple, until one rule fits all three. The outline shape and the total number of dots do not matter.
Know first: Counting small groups quickly, Doubling and adding small numbers
Step 1 of 5
The rule links two counts
Some classification questions show a large outline, such as a circle, triangle or square, with dots inside it and dots outside it. The outline changes from example to example, and so do the dot positions. What stays the same is how the inside count compares with the outside count.
Write each example as a pair, like 4 in, 2 out. Then test rules until one fits all three examples.
| Rule | Inside count when 3 are outside | Quick test |
|---|---|---|
| Equal | 3 | Inside minus outside is 0 every time |
| Plus 1 or plus 2 | 4 or 5 | Inside minus outside is the same every time |
| Minus 1 | 2 | Outside minus inside is 1 every time |
| Twice | 6 | Half the inside count equals the outside count |
| Twice plus 1 | 7 | Double the outside count, then add 1 |
Test the difference first. If the difference changes, try doubling.
Step 2 of 5
Line up the pairs and test
Worked example: Find the rule, then the match
Example 1: a circle with 4 dots inside and 2 outside.
Example 2: a triangle with 6 inside and 3 outside.
Example 3: a square with 2 inside and 1 outside.
Which choice belongs with them?
- APentagon, 5 inside, 3 outside
- BCircle, 4 inside, 4 outside
- CHexagon, 8 inside, 4 outside
- DSquare, 6 inside, 2 outside
- ETriangle, 3 inside, 6 outside
- 1
Test the difference
4 − 2 = 2, 6 − 3 = 3, 2 − 1 = 1. The difference changes, so no "plus" rule works.
- 2
Test doubling
2 × 2 = 4, 2 × 3 = 6, 2 × 1 = 2. All three fit. Inside is twice outside.
- 3
Test the choices
Pentagon 5 in, 3 out: 2 × 3 = 6, not 5. Circle 4 and 4: equal. Hexagon 8 in, 4 out: 2 × 4 = 8, it fits. Square 6 in, 2 out: three times. Triangle 3 in, 6 out: reversed.
- 4
See why the other choices are there
The circle copies Example 1's outline. The triangle has 9 dots in all, the same total as Example 2, and flips which count is bigger. The square is a multiple, just the wrong one.
Answer
Hexagon, 8 inside, 4 outside.
Wrong: The circle choice matches Example 1, so it belongs.
Right: The hexagon choice matches the counting rule, so it belongs.
The outlines change on purpose. Only the inside-to-outside relationship carries over.
Step 3 of 5
Find and apply the rule
Check yourself · Question 1
Example 1: 3 dots inside, 1 outside. Example 2: 5 inside, 3 outside. Example 3: 4 inside, 2 outside. Which rule fits all three?
Answer: C
The difference is 2 every time: 3 − 1, 5 − 3 and 4 − 2.
Check yourself · Question 2
Example 1: a triangle with 3 dots inside and 1 outside. Example 2: a circle with 5 inside and 2 outside. Example 3: a hexagon with 7 inside and 3 outside. Which choice belongs with them?
Answer: C
Doubling alone gives 2, 4 and 6, one short each time. The rule is twice the outside count plus 1: 3, 5, 7. For 4 outside, that is 2 × 4 + 1 = 9 inside.
Trap answer: Square, 8 inside, 4 outside
Half the rule
- Why it looks right
- Doubling is the first multiple to try, and 8 and 4 look like a clean match.
- Why it's wrong
- Twice outside gives 2, 4 and 6 for the examples, but they have 3, 5 and 7 inside. Every example has one extra.
- The right answer
- Square, 9 inside, 4 outside, since 2 × 4 + 1 = 9.
Common mistake
I counted all the dots together.
- Why it's tempting
- The total is one number, and it is faster than counting two groups.
- Do this instead
- The rule compares two groups. Decide for each dot whether it is inside or outside the outline, then count the groups separately.
Step 4 of 5
What to remember
Remember
- Write each example as a pair: inside count, outside count.
- Test the difference first, then doubling, and keep only a rule that fits all three examples.
- Ignore the outline shape and the total number of dots.
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 question, what inside and outside counts did you write for each example, and what rule fit all three?