|Time and Space|
The answers to these problems are actually analogous to those of the two jugs problem, although there are exceptions. This is of course because time can be ‘banked’ having been measured in this problem, whereas in the jugs problem all the water must stay within the jugs.
That been said you can of course measure a greater amount of time than the sum of the two sand glasses you are given; rather than the less than or equal restraint that governs the jugs.
In the water jugs puzzles we learned that the key move is when you have more water in one jug than available space in the other jug. Pouring between the two then gives you a split of known quantity.
Here the trick is running both timers from the start and letting them get out of sync, due to their different run times. Then you can time from the end of one timer to the end of the other timer. Here’s an example to look at.
How can you measure four minutes with a sand timer measuring 3 minutes, and another measuring 5 minutes?
If we were to solve this mathematically we’d be looking for a solution to the following equation.
5x – 3y = |4|
x is the number of turns for the larger timer, y the number of turns for the smaller; and we multiply that by the duration of the timer, to give the time they’ve timed so far. The bars around the four are modulus signs. They mean we don’t mind if the answer is positive or negative.
The trouble here is a linear equation in x and y is hard to solve except by trial and error; and there’s a much better way of solving the problem, what we need is visualisation.
Wow, here's something crazy, you can watch a video explaining how to construct the visualisation for this problem. Or if you prefer you can skip onto to the text based version.
So, to summarise what we went over in the video, you can use timelines to help visualise the sand timers puzzle - even if the puzzle is in minutes and the timelines are in seconds - ahem.
Construct the timeline by plotting all the turns of one timer on the top, and all the turns of the other timer on the bottom. Then combine the two so that you have a timeline showing all available turn points; these are of course the points where we know exactly how long has passed.
This timeline then, gives a set of time-points which in turn give a series of durations that can be timed. Also don't forget that by shifting the those points along the timeline by the duration of one of the timers, both of the timers, or one timer multiple times you also increase the durations you can measure.
Here's the original timeline, this time in fancy graphical form showing the times that can be made without staggering.
Bonus points incidentally if you spotted my mistake in the video - it's perfectly possible to time seven units on the original timeline, Eight is the first number that causes a problem and since the solution for that is in the video (just stagger the time line by the duration of the five minute timer) - here are my challenges for you.
How can you measure eleven minutes with a sand timer measuring 3 minutes, and another measuring 5 minutes?
What is the first round minute duration that cannot be measured with a 4 minute timer and a 7 minute timer if you have to start the timers together and turn them every time they run out
How could you measure that duration with the two timers?
Are there any round-minute durations that cannot be measured?
Why can't they? or possibly Why not?
That's all from me for today - happy puzzling!