Problem 19 · 2016 Math Kangaroo
Hard
Number Theory
caseworkdivisibility
Each of the ten points in the diagram is labelled with one of the numbers 0, 1 or 2. It is known that the sum of the numbers in the corner points of each white triangle is divisible by 3, while the sum of the numbers in the corner points of each black triangle is not divisible by 3. Three of the points are already labelled as shown in the diagram. With which numbers can the inner point be labelled?
Show answer
Answer: A — only 0
Show hints
Hint 1 of 3
Read every condition modulo 3: a white triangle's three corners sum to 0, a black triangle's do not.
Still stuck? Show hint 2 →
Hint 2 of 3
Two triangles that share an edge differ only in their third corner, so compare the apex labels of triangles sitting on the same base.
Still stuck? Show hint 3 →
Hint 3 of 3
Chase the conditions outward from the given 0 and 2 to pin the inner point's residue.
Show solution
Approach: residues mod 3 on the triangular grid
- Work mod 3: each white (upward) triangle's corners sum to 0, each black (downward) triangle's corners do not.
- An upward and the downward triangle resting on the same edge share two corners, so their third corners must have different residues; this forces neighbouring apex labels apart.
- Starting from the labelled corners 0 and 2 and propagating these forced differences leaves the inner point only one possible residue.
- That residue is 0, so the inner point can be only 0 (A).
Mark:
· log in to save