November 2004
Shapes
The diagram shows how a 7 by 7 square can be divided into 9 smaller squares. Can you show how a 13 by 13 square can be divided into 11 smaller squares in a similar way?
Extension
What is the least number of smaller squares which a 12 by 12 square can be divided into? What about a 11 by 11 square? Or a 10 by 10 square, etc.? (Some cases are very obvious, others less so.)
Numbers
The diagram shows a grid which contains 6 sets of 4 numbers:
- a set of 4 square numbers
- a set of 4 triangular numbers
- a set of 4 prime numbers
- a set of 4 Fibonacci numbers
- a set of 4 odd numbers
- a set of 4 factors of 60
No two members of a set are in the same row or column. So, for example, 25 and 36 cannot both be in the set of 4 square numbers.
Can you work out which numbers are in each of the 6 sets, and therefore which number is left over at the end, i.e. is not in any of the sets?
Try to explain the logic that you use, at least for the first few numbers that you place into sets.
Algebra
Alan throws darts at a dartboard and hits 3 of the numbers from 1 to 20. The total of these 3 numbers is 31. However, one of the darts lands in the ‘treble’ section, and one lands in the ‘double’ section, whilst the third lands in the ‘single’ section, so his overall score for the 3 darts is 64.
Prove that the number of the dart which lands in the ‘treble’ section is 2 more than the number of the dart which lands in the ‘single’ section.
Miscellaneous
The diagram shows 24 coins which have been placed in 3 piles. The first pile contains 6 coins, the second pile contains 11 coins, and the third pile contains 7 coins.
A ‘move’ consists of transferring coins from one of the piles to another pile; the number of coins transferred must be equal to the number of coins already in the pile to which the coins are being transferred. So, for example, a possible starting move would be to transfer 6 coins from the ‘11’ pile to the ‘6’ pile, giving piles with 12 coins, 5 coins, and 7 coins.
The objective is to make all of the piles equal, with 8 coins in each.
Show how to do this in the least number of moves.
Extension
Is it possible to place the 24 coins in a starting arrangement which makes the puzzle impossible? If not, what starting arrangement takes the greatest number of moves?