Building a new track

Number Theory Level 1

At an athletics hall, there is usually a track for the 60-meter dash. It also has a starting line for a 50-meter dash with respect to the same finish line.

In training, we wanted to run 30 meters, but there was no line for that.

What is the minimum number of additional lines that need to be added so that every distance { 10 , 20 , 30 , 40 , 50 , 60 } \{ 10,20,30,40,50,60 \} meters can be run on this track?

Notes:

  • The distances don't have to end at the usual finish line; they can also take any other line as start or finish.
  • All lines have to be between the 60-meter start and finish line.


The answer is 1.

Henry U by Henry U , 17, Germany

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Roger Erisman
Dec 27, 2018

Lines already at Start, 10, 60

Add line at 40

10 Start to 10

20 40 to 60

30 10 to 40

40 Start to 40

50 10 to 60

60 Start to 60

17 Helpful 6 Interesting 9 Brilliant 1 Confused
Cheuk Yin Kwan
Dec 31, 2018

We have one line at the 60m mark and another one at the 50m mark. We just need to add one more line at the 20m mark. That way,

  • if we want to run 10m, just run from the 60m line to the 50m line;

  • if we want to run 20m, just run from the 20m line to the finish line;

  • if we want to run 30m, just run from the 50m line to the 20m line;

  • if we want to run 40m, just run from the 60m line to the 20m line;

  • if we want to run 50m, just run from the 50m line to the finish line;

  • if we want to run 60m, just run from the 60m line to the finish line.

2 Helpful 1 Interesting 0 Brilliant 1 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...