Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some other
possibilities for yourself!
Can you explain the strategy for winning this game with any target?
Matthew Turner (Queen Mary's Grammar School) sent us a lovely
solution to this problem.
You could work with 999 as well as 1000 as the question says
"numbers less than 1000" but the solution would have been the same
Here is Matthew's solution:
By dividing 100 by five I know there are 200 multiples of
By dividing by seven I know there are 142 multiples of
To work out the how many multiples of five, two and seven, we
have to find the LCM of them, which is 70. This goes into 1000
fourteen times, so 14 goes into the area where all three circles
There are 100 numbers that are multiples of two and five (LCM
10) less the 14 that are also multiples of 7, leaves 86.
There are 71 numbers that are multiples of two and seven (LCM
14) less the 14 that are also multiples of 5, leaves 57.
There are 28 numbers that are multiples of seven and five (LCM
35) less the 14 that are also multiples of 2, leaves 14.
To work out how much was in the biggest parts of the circles I
added up the other three parts and subtracted each of these figures
from the total number of multiples of each of the numbers.
e.g. for two
57 + 86 + 14 = 157
500 - 157 = 343
Then we add up every number within the three circles which adds
up to 657 and take that from 1000, which is 343.
For the answer to the first part we only need to consider the
multiples of 2 and 5 so
343 + 57 + 86 + 14 + 86 + 14 = 600
1000 - 600 is 400.