We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
Can you make sense of the three methods to work out the area of the kite in the square?
Biren Patel, Heathland School, Hounslow, England; Andrei, School
no. 205, Bucharest, Romania and Ang Zhi and Chai from River Valley
High School, Singapore sent in very good solutions. Here is Chai
Prove that if n is a triangular number then 8n+1 is a square
Prove, conversely, that if 8n+1 is a square number then n is a
Substitute n = K/2 (K+1) into 8n + 1
Therefore, if n is a triangular number then 8n+1 is a square
To prove the converse, let X 2 = 8n + 1 be a square
As 8 is an even number, 8n will always be an even number. If 8n
is an even number, then 8n + 1 will always be an odd number. X
cannot be even because the square of an even number is even.
Hence X= (2k + 1) represents an odd number where k can be any
(2k + 1) 2 = 8n + 1
4k 2 + 4k+ 1 = 8n + 1
4k 2 + 4k + 1 - 1 = 8n
4/8(k 2 + k) = n
(k 2 + k)/2 = n
Therefore, n = k/2 (k+1) so n is a triangular number.