### Pinned Squares

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a 5 by 5 board?

### Icosagram

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at the number of diagonals?

### AP Rectangles

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

# Hot Pursuit

##### Stage: 3 Challenge Level:

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed? Some people just added up 1+2+3+ etc. until they found the first total in which all the digits are the same. The sum of the first 36 numbers from 1 to 36 add up to 666 so the answer must be that 36 numbers have been summed. There are better methods.

Soh Yong Sheng, 13, Raffles Institution, Singapore recognsied that if the sum has all the digits the same it must be a multiple of 111 and used the fact that the sum of the first n whole number is n(n+1)/2

Since the sum has to be a three digit number in which all the digits are equal, the sum has to be a multiple of 111 = 37 x 3.

Hence n or n+1 is a multiple of 37. But the product is a three digit number and hence higher multiples can be ignored. Therefore n or n+1 is 37.

If n = 37 then n +1 = 38 and the product is not a multiple of 3. So n+1 = 37 and n = 36 .

Soh Yong Sheng also used this method:

The question gives an equation 111a = n(n+1)/2 . Thus we need to find n where 222a = n 2 + n, 1

Listing the possibilities:

a=1: 14 2 + 14 < 222 and 15 2 + 15 > 222
a=2: 20 2 + 20 < 444 and 21 2 + 22 > 444
a=3: 25 2 + 25 < 666 and 26 2 + 26 > 666
a=4: 29 2 + 29 < 888 and 30 2 + 30 > 888
a=5: 32 2 + 32 < 1110 and 33 2 + 33 > 1110
a=6: 36 2 + 36 = 1332
Hence n = 36