### Patio

A square patio was tiled with square tiles all the same size. Some of the tiles were removed from the middle of the patio in order to make a square flower bed, but the number of the remaining tiles was still a square number. What were the dimensions of the patio and the flower bed?

### Time of Birth

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

### Square Routes

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

# Digat

##### Stage: 3 Challenge Level:

Congratulations to the Key Stage 3 Maths Club at Strabane Grammar School. We give below their methods of finding A where [3(230 + A)] 2 = 492 80A.

The number on the right must be divisible by 9 because of the 3 which is squared on the left. This means its digits must add to give a multiple of 9 - (9, 18, 27, 36,...... etc).

4+9+2+8 = 23
23 + A must give 27 or 36 or 45 etc.
27 is the only total we can reach since A is a digit so A = 4.
We also wrote a program in Basic to check this.
 > 10 FOR A = 0 TO 9 > 20 IF ((230 + A)*3)^2 = 492800 + A PRINT A > 30 NEXT A Escape > RUN 4

Elizabeth of Ipswich High School checked the square roots of 49280A to find which one was an integer. James of Hethersett High School, and S.Kuo of Eaton School Norwich, and groups from Ousedale School, Newport Pagnell and Madras College, St Andrew's and Bithian and Guobin from The Chinese High School, Singapore, used mixtures of logical deduction, algebra and trial and error, homing in between answers that turned out too low and too high.