Count the number of squares in the given figure.
The figure may be labelled as shown.
The simplest squares are QUYX, URVY, YVSW and XYWT i.e. 4 in number.
The squares composed of two components each are IMYP, MJNY, YNKO and PYOL i.e. 4 in number.
The squares composed of three components each are AEYH, EBFY, YFCG and HYGD i.e. 4 in number.
There is only one square i.e. QRST composed of four components.
There is only one square i.e. IJKL composed of eight components.
There is only one square i.e. ABCD composed of twelve components.
Total number of squares in the given figure = 4 + 4 + 4+1 + 1 + 1 = 15.
What is the minimum number of colours required to fill the spaces in the given diagram without any two adjacent spaces having the same colour?
The figure may be labelled as shown.
The spaces P, Q and R have to be shaded by three different colours definitely (since each of these three spaces lies adjacent to the other two).
Now, in order that no two adjacent spaces be shaded by the same colour, the spaces T, U and S must be shaded with the colours of the spaces P, Q and R respectively.
Also the spaces X, V and W must be shaded with the colours of the spaces S, T and U respectively i.e. with the colours of the spaces R, P and Q respectively. Thus, minimum three colours are required.