Building a new track and runnig – 5

Number Theory Level 2

Now, at the 400m-track, I want to do the following runs in training:

$\{50, 100, 150, 200, 150, 100, 50\}$

After every run, I stop directly and stay at this line. The next run will start from this same line.

The track has lines at the following distances relative to the finish:

$\{0, 50, 100, 150, 200, 250, 300\}$

At which line do I have to start to do my training in the described way?

Example:

I can't start at the 0m-line because then, I don't know where to stop after 50m.

150 200 50 100 300 250 0

1 solution

Henry U
Dec 29, 2018

We measure everything in units of 50m.

During the workout, I will run the following "partial sum" distanes

$\{1,3,6,10,13,15,16\}$ .

Since the track is circular with a length of 8, we have to compute the remainder of these numbers under division by 8

$\{1,3,6,2,5,7,0\}$

Note that every possible remainder occurs, except 4. On the track, this means that we will get to every 50m-line exactly once, but not to the one that is 200m away from the start. This fits perfectly with the condition that there is no 350m-line, so we have to start such that the place, where this line would be, is 200m away from the start of our run. The only possible starting point for this is at 150m .

Building a new track and runnig – 5

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