Every card in IMC 2026

2

HintTwo staircases make a rectangle.

WhyTriangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91. The mean of 1 to n is (n + 1) ÷ 2, so those means grow by a half each step.

It factorises as (n − 1)(n + 1)

HintWhat is 1 the square of?

Why(n − 1)n(n + 1) is a product of three consecutive numbers, so it is always a multiple of 6; for n not divisible by 2 or 3, (n − 1)(n + 1) is a multiple of 8 and of 3, hence of 24 — which is why 17, 19 and 23 work while 21 and 27 do not.

Clear the fractions first

HintA ratio survives one particular operation on every part.

Why1 : ½ : ⅓ is 6 : 3 : 2 (multiply by 6), eleven parts in all. A ratio is unchanged when every part is multiplied by the same number — so fractions and decimals never need to stay in it.

Equal to the rectangle's

HintSlide every inward edge outwards.

WhyEach horizontal edge projects onto the top or bottom of the enclosing rectangle, each vertical edge onto a side, with nothing double-counted — so a staircase shape has perimeter 2(width + height). Add any sloping edges separately.

a√2

HintHalf a square, cut corner to corner.

WhySo on a figure whose angles are all multiples of 45°, a sloping edge is √2 times its horizontal (= vertical) span, and two sloping edges that together rise 3 have total length 3√2. Split the perimeter into level, upright and sloping parts.

(√3 ÷ 4) s²

HintHalf the base times a height found by Pythagoras.

WhyHeight = (√3 ÷ 2) s, so area = ½ × s × (√3 ÷ 2) s. A parallelogram's area is base × perpendicular height — never base × slant side. Ratios of areas built from the same √3 cancel it, leaving a whole-number ratio.

Two equal halves

HintWhat does the line change about the two pieces, and what does it not?

WhyEqual bases plus a shared perpendicular height means equal areas. Two such lines from two vertices split the triangle into four pieces you can pair up.

3xy

HintExpand the cube and collect the mixed terms.

WhyWith x + y = 5 and x³ + y³ = 95: 125 = 95 + 15xy, so xy = 2 — no need to find x or y. Its square cousin: (x + y)² = x² + y² + 2xy.

100

HintWrite the percentage as a fraction in lowest terms and look at the bottom.

Why97/100 × N is a whole number only when 100 divides N, because 97 and 100 are coprime. 90% is different: 9/10 × N only needs N to be a multiple of 10. Reduce the percentage to lowest terms and read the denominator.

Division by the number of orders

HintThree items can be lined up in how many ways?

WhyPicking three things in order gives n × (n − 1) × (n − 2) sequences, and each set of three appears in 3 × 2 × 1 = 6 of them, so divide by 6. Then count the favourable sets systematically and divide.

1; even

HintLargest digit plus largest digit plus a carry — how big can that be?

Why9 + 9 + 1 = 19, so a carry is never more than 1; a column total of 2A or 2A + 1 tells you at once whether a carry came in; a leading digit is never 0. Three facts crack most letter-sum puzzles.

Peel from the outside inwards

HintWhich layer was built last?

WhyMove the outer constant, take reciprocals, and the next layer appears; three or four repeats reach x. Simplifying (rather than solving) goes the other way, from the innermost fraction out.