Hidden Faces The student’s professor is upside-down. The girl’s chin is the grandmother’s nose.
Pages 10 and 11/Knight's puzzle
The knight can make it in 11 moves. There are other, longer routes.
The numbers in the squares show the order of one of the ways they can all be reached using knight’s moves starting at 1. (This is hard to annotate as the knight has to go back many times to check all the squares he can reach from each position - can a more economical route be found?)
Pages 12 and 13 /Remy paints a cycle
3: one cycle 1,2 7: one cycle 3,4 1,6 5,2 9: two cycles 4,5 1,8 7,2 and 3,6
11: one cycle 5,6 1,10 9,2 7,4 3,8 15: See below
23: one cycle 12,11 1,22 21,2 19,4 15,8 7,16 9,14 5,18 13,10 3,20 17,6
Children will recognise that each pair of numbers created by the rules sums to the number being cycled.
For able children this can be demonstrated and justified algebraically.
When the starting number is a prime number the cycle contains all the possible combinations of pairs of
numbers that sum to the starting number (with the exception of 17 are there others?).
Other odd numbers may need more that one cycle to exhibit all the possible pairs. For example, 15 has
3 cycles of pairs:
For even numbers the pairs form loops rather than cycles. Usually, the separate loops are linked and
have a final pair which includes zero.
Remy’s cycle begins an investigation in which a variety of questions can provide some fruitful starting
points. For example children can categorise* the types of number which create cycles, loops and spurs.
Pages 14 and 15 /Are you puzzled?
Augusta sold two paintings at £70.00 and five at £20.00.
The third painting will have 24 white cells.
The tenth painting will have 220 white cells.
For any n x n grid, excluding the centre red cell, there are an equal number of red and white cells.
So the number of white cells for an n x n grid is 1/2(n2-1).
Who stole the Haywain?
Mr Brush stole the Haywain.
Page 16 (back cover)/How much?
In frame 4 the bottom set contains 1 more item than the set above. By subtracting the price of the top
set from the bottom set you get the price of the milk jug. The prices of the other items (even the price
of the individual mugs) can be found in a similar way.
The teapot is £7.00, the 4 mugs are £5.00 (£1.25p each), the sugar bowl is £3.00, the milk jug is £3.50
and the cake stand is £4.00.