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Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Quite a few students wrote in with correct solutions and two groups offered short, precise reasoning.

The first group of four: Luke, Dan, Nicholas and Luke, all from Clevedon Community School , point out that:

Any number of minutes are possible provided it is possible to time 1 minute.

While the second group of Jessica, with the help from Gemma and Tim describe, quite nicely, how to time one minute:

1 minute can be created by turning the 4 minute timer (4m) and the 7 minute timer (7m) over then as the 4m runs out you turn it over and when the 7m runs out you have one minute. Therefore, you can repeat this process x no. of times (leaving out the gaps turning the 7m over) making any number x.

Edmund, The Chinese School in Singapore; Claire, Madras College in St. Andrews; and, Jimmy, West Flegg Middle School in Great Yarmouth, also correctly solved the problem . Well done.

However, Peter wrote to us saying:

Simply repeating the one minute from end-of-T7 to the second end-of-T4 will not work as you will have to wait another $7$ minutes between the first minute and th second.

To get contiguous minutes (i.e. minutes which directly follow on from each other) you need to subtract multiples of $4$ and $7$:

$1=8-7$

$2=14-12$

$3=7-4$

$4=4$

$5=12-7$

$6=14-8$

$7=7$

$8=8$

$9=21-12$

$10=14-4$.

The exception is $11$ where you add the two together.

Thank you for this clear solution, Peter.