Two egg timers
Problem
You have two egg timers.
One takes 4 minutes exactly to empty and the other takes 7 minutes.
What times in whole minutes can you measure and how? (You can ignore the time it takes to reset the timers.)
Getting Started
Do you have a system for trying out your ideas?
Why would it help to have a way of making 1 minute?
Student Solutions
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.