Every card in IMC 2025
The whole deck, in order — so you can read it through before your child ever sees it.
- The units digits of the powers of 8 run 8, 4, 2, ____ and then repeat, a cycle of four.
6
HintWhat does 2 × 8 end in?
WhyPowers of 2, 3, 7 and 8 all have units-digit cycles of length 4; 4 and 9 have length 2; 0, 1, 5, 6 never change. The remainder on dividing by 5 (or 10) is read straight from that units digit — 8¹⁰ ends in 4, so it leaves remainder 4 on division by 5.
- The exterior angle of a triangle
The sum of the two opposite interior angles
HintCombine the straight line with the triangle's own total.
WhyIf the base angles are b, the exterior angle at the base is 180 − b and also equals apex + b. Isosceles triangles hand you two equal base angles for free — say so before you write anything else.
- Every item costs 1p less than a whole number of pounds. What does the total's shortfall from a whole number of pounds tell you?
The number of items
HintThink of paying whole pounds and getting change.
WhyPrices like 99p and £2.99 are whole pounds minus a penny, so 25 such items cost 25p less than a whole number of pounds. Read the count off the pence, then a second equation on the pounds finds how many of each.
- Adding several single-digit recurring decimals — what do you add?
Ninths
Hint0.1 recurring is 1 over what?
WhyA single repeating digit d is d/9: 0.1 recurring = 1/9, 0.6 recurring = 2/3, 0.9 recurring = 1. So a sum of them is a fraction over 9, which may itself be a recurring decimal.
- (a + 1)(b + 1) − ab = a + b + 1, so a product and its 'add one to each' partner give the ____ at once.
sum
HintNot the numbers themselves.
WhyExpand the brackets, then substitute the given product rather than solving for a and b — most 'find the sum' questions never need the numbers themselves.
- A point P sits inside a rectangle. What links its distances to the four corners?
Opposite corners: equal sums of squares
HintPythagoras four times, then pair up.
WhyDropping perpendiculars from P to the sides makes four right-angled triangles sharing legs; adding the two 'opposite' Pythagoras equations gives PA² + PC² = PB² + PD². Distances 6, 8 and 10 to three corners in order force the fourth to be √(36 + 100 − 64) = √72.
- Two circles of radii r and R touch each other from the outside. What is the distance between their centres?
r + R
HintThe touching point lies on the line of centres.
WhyCentres and touching point are collinear, so the centre distance is r + R when the circles touch from outside and R − r when one sits inside the other.
- A class has three times as many girls as boys. What must the class size be?
A multiple of 4
HintHow many equal shares make the whole?
WhyA ratio a : b in whole parts makes the total a multiple of a + b: 3 : 2 → multiple of 5, 5 : 1 → multiple of 6, 7 : 3 → multiple of 10. Matching totals to ratios is then a divisibility check, not an equation.
- Writing a 3 : 5 ratio as actual amounts with one unknown
3n and 5n
HintOne shared multiplier scales both parts.
WhyIntroduce one letter for the common multiplier: 3n red and 5n blue counters; add 4 red and the ratio is 1 : 1, so 3n + 4 = 5n and n = 2.
- For whole-number n, the expression 2n is always ____, 2n + 1 is always ____, and n(n + 1) is always a multiple of ____.
even; odd; 2
HintWhich multipliers keep the parity and which force it?
WhySo 5n + 3 even forces n odd; then 7n + 2 is odd and 4n + 1 is odd for every n. Parity questions are settled by checking n = 1 and n = 2 — nothing else changes the pattern.
- (5⁶ − 5⁴) ÷ (2⁸ − 2⁶): what is the first move?
Factorise each bracket
HintBoth brackets share a common power.
Why5⁴(5² − 1) ÷ 2⁶(2² − 1) = 625 × 24 ÷ (64 × 3) = 625 × 8 ÷ 64 = 625/8. Never expand a power you can factorise.
- The number of ways to choose 2 items from n is n(n − 1) divided by…? (type the number only)
2
HintEach pair got counted in both orders.
WhyChoosing 2 from 6 gives 6 × 5 ÷ 2 = 15, from 7 it is 21. Constraints such as 'not next to each other' are handled by listing by the first chosen item and counting what remains legal.
1★ KS3-MATH-IMC-0133
2★ KS3-MATH-IMC-0134
3★ KS3-MATH-IMC-0135
4★ KS3-MATH-IMC-0136
5★ KS3-MATH-IMC-0137
6★ KS3-MATH-IMC-0138
7★ KS3-MATH-IMC-0139
8★ KS3-MATH-IMC-0140
9★ KS3-MATH-IMC-0141
10★ KS3-MATH-IMC-0142
11★ KS3-MATH-IMC-0143
12★ KS3-MATH-IMC-0144
Keep what you learn
Here, nothing is saved. In your child’s own sky every card is scheduled — it comes back just before they’d forget it — and the professor who wrote it is one tap away.