Prize Puzzle Competition 20

Closing date is 30th December 1999

58. It is possible to use all the whole numbers from 1 to 16 and match them up into eight pairs so that each pair adds up to a square number.
For example, (1, 3) and (2, 14) could be two of the pairs since they give 4 and 16, both of which are square numbers.
List all the eight pairs.

59. The numbers 1 to 12 have been written on four lines as shown.
There is a rule which fixes on which line each number must go.







Keeping to that same rule
where must 13 go?

60. Tom has a contract to dig out some foundations and it must be done in 30 days.
His own machine, which he wishes to use as much as possible, would take 48 days to do all the work. He can hire a bigger machine which would do the complete job in 21 days, but it costs 300 a day.
There is only enough room for one machine at a time.
What is the least number of days for which he will have to hire the larger machine?

The winner of Competition #19 was Sean Doherty, St. Columb's College, Londonderry.
The answers were:

55.All conditions can be met with 1, 3, 5 or 7 - 50p pieces.
56.It could only be you.
However, 'brother or sister' was also allowed since, in the first version of the puzzle, the 3rd 'only' had been left out.
57. The height was 2.49 metres (to 2 d.p.).