Singapore Math Drills Team · 7 November 2026
Free, distraction-free math drills aligned to the MOE syllabus. No sign-up needed to try.
Every first-contact film needs a listening station. This one is a bedroom: a tinfoil salad-bowl dish on a broom handle, aimed at the star stickers on the ceiling, and a shoebox receiver with five signal bars. The Mastermind, a tinfoil colander over his red beanie, turns the knob: "We are not alone." His plan: "Five questions a day." The Partner, juice box in hand: "...Five?"
That is Message From the Stars, episode 10 of our comedy series The Mastermind and our first Secondary 4 episode. Watch it first (under a minute).
Each question slip from the receiver gets an "Incoming." and lights one more signal bar. On the fifth, the message unrolls: a printout of 12 doodled aliens. 7 have antennae, 5 have three eyes and 2 have both. Below them is a Venn diagram with a 2 already written in the overlap. The Partner's verdict: "They came in peace. For practice."
That first move, overlap first, is the habit that makes Venn diagram questions calm.
A two-set Venn diagram looks simple: a rectangle for everything, two overlapping circles inside. The trouble is that "7 have antennae" counts every alien in the antennae circle, including those in the overlap. Write 7 in the antennae-only part and the overlap is counted twice, so every number after it is wrong.
In the MOE teaching sequence, sets and Venn diagrams belong to upper-secondary G3 Mathematics, in Sec 3 and Sec 4. Not sure which level your child takes? See our guide to G1, G2 and G3 Mathematics.
The rectangle ξ is the universal set. Inside it are four regions: A only, both (A ∩ B, the intersection), B only, and neither. A ∪ B (the union) means "in at least one". Fill the regions in this order:
n(A ∩ B), in the overlap.n(A) − n(A ∩ B), and B only is n(B) − n(A ∩ B).n(A ∪ B), and it should equal n(A) + n(B) − n(A ∩ B).n(ξ) − n(A ∪ B).Of 12 aliens, 7 have antennae, 5 have three eyes and 2 have both. How many aliens have neither antennae nor three eyes?
Step 1: Both first. Write 2 in the overlap.
Step 2: Each circle. Antennae only: 7 − 2 = 5. Three eyes only: 5 − 2 = 3.
Step 3: At least one. 5 + 2 + 3 = 10, which matches 7 + 5 − 2 = 10 ✓.
Step 4: Neither. 12 − 10 = 2, the two plain aliens outside both circles.
Step 5: Check. 5 + 2 + 3 + 2 = 12 ✓. 2 aliens have neither.
| Region | Aliens |
|---|---|
| Antennae only | 5 |
| Both | 2 |
| Three eyes only | 3 |
| Neither | 2 |
| Total | 12 |
Notice the two 5s: antennae only is 7 − 2, not the 5 three-eyed aliens. Label each region as you fill it.
7 + 2 + 5 = 14, more than the 12 aliens on the printout. The 2 with both were already inside the 7 and the 5.7 + 5 = 12 makes it look as if every alien has a feature, so "neither" comes out as 0. Take the overlap away once: 7 + 5 − 2 = 10.One house rule for every two-set Venn diagram: when it is finished, add the four regions and compare with the total. If they do not match, the overlap has usually been counted twice or left out.
Build a Venn diagram on the kitchen table with two loops of string or two plates. Pick two labels, such as "wears glasses" and "likes durian", and let your child sort the family into the regions. Then ask the other way round: "How many like durian? How many are in both? In neither?" Placing people by hand shows that anyone in the overlap belongs to both circles.
Keep it light. When your child is stuck, ask "What goes in the overlap?" before "What is the answer?", and praise the order of working, not the speed.
Overlap first, then the four-region check.
Question 1. In a group of 20 students, 12 have a cat, 9 have a dog and 4 have both. How many have neither a cat nor a dog?
Both 4. Cat only 12 − 4 = 8, dog only 9 − 4 = 5. At least one: 8 + 4 + 5 = 17. Neither: 20 − 17 = 3. Check: 8 + 4 + 5 + 3 = 20 ✓.
Answer: 3 students have neither.
Question 2. Of 25 students, 15 take music, 11 take art and 6 take both. How many take exactly one subject, and how many take neither?
Both 6. Music only 15 − 6 = 9, art only 11 − 6 = 5. Exactly one: 9 + 5 = 14. Neither: 25 − (9 + 6 + 5) = 5. Check: 9 + 6 + 5 + 5 = 25 ✓.
Answer: 14 take exactly one subject, and 5 take neither.
Question 3. In a class of 30, 14 are in the chess club, 10 are in the robotics club and 9 are in neither. How many are in both clubs?
This one runs backwards. At least one: 30 − 9 = 21, so 14 + 10 − n(both) = 21 and n(both) = 3. Chess only 14 − 3 = 11, robotics only 10 − 3 = 7. Check: 11 + 3 + 7 + 9 = 30 ✓.
Answer: 3 are in both clubs.
Question 4. ξ = {1, 2, 3, …, 12}, A is the set of even numbers and B the set of multiples of 3. Find A ∩ B, n(A ∪ B) and the elements in neither set.
A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}. Both first: A ∩ B = {6, 12}. A only {2, 4, 8, 10}, B only {3, 9}, so n(A ∪ B) = 4 + 2 + 2 = 8. Neither: {1, 5, 7, 11}. Check: 4 + 2 + 2 + 4 = 12 ✓.
Answer: A ∩ B = {6, 12}, n(A ∪ B) = 8; neither: 1, 5, 7 and 11.
In the episode, the day counter goes Day 1, Day 7, Day 30, and it only ever goes up. Venn diagrams become comfortable the same way: one habit, met often, checked every time.
That is the idea we built Singapore Math Drills around:
Sets & Venn Diagrams is one of six selected G3-tagged topics we currently offer for Secondary 4, alongside Vectors and Probability, the other two constellations on the episode's star chart. Our Secondary questions do not cover the complete syllabus or a specific track; they are practice for the topics we list.
The Mastermind keeps his eyes on the dish, all five bars still lit. Told the visitors only came for practice, he whispers: "Earth must never know."
Try a free interactive Secondary 4 worksheet, with no signup or credit card required.