Problem 15 · 2023 Math Kangaroo
Medium
Counting & Probability
caseworkcareful-counting
A big rectangle is made up of five small rectangles (see picture). Lukas wants to colour the small rectangles red, blue and yellow so that any two rectangles sharing a side have different colours. In how many ways can he do this?
Show answer
Answer: D — 6
Show hints
Hint 1 of 3
Start by colouring one rectangle, then move to a neighbour and count how many of the three colours are still allowed.
Still stuck? Show hint 2 →
Hint 2 of 3
Multiply the number of free choices as you colour the rectangles one by one.
Still stuck? Show hint 3 →
Hint 3 of 3
Be extra careful at a rectangle that touches two already-coloured neighbours, since it may have fewer choices left.
Show solution
Approach: colour the rectangles in order, counting choices for each
- Colour the three top rectangles left to right: the first has 3 choices, each next one must differ from its neighbour.
- The bottom rectangles must differ from the top rectangles they touch and from each other, which limits the remaining choices.
- Carefully multiplying the allowed choices through the whole arrangement gives 6 valid colourings.
- The answer is D, 6.
Mark:
· log in to save