I copied this question from How are the pigeonholes calculated in this pigeon-hole problem?

To prepare for a marathon, an elite runner runs at least once a day over the next 44 days, for a total of 70 runs in all. Show that there's a period of consecutive days during which the runner runs exactly 17 times.

I tried. But I can't understand this problem. How is it possible that there is a consecutive period of days during the runner runs exactly 17 times?

Here are my inferences:

  1. Runner runs daily. So each day the number of runs increases by 1 (at least)
  2. If he completed 15 runs in someday, the next day it will be 16, then 17 and so on.. (at least, because he runs daily)
  3. Then where is the period of consecutive days? I am lost here!

Mathematically it can be shown as mentioned in the linked question, but I can't understand the logic behind it. Any help would be appreciated. Thanks!

  • 1
    $\begingroup$ Your question is missing some important context: What about the solution already given in the linked question do you not understand? You should explain this, telling us what happened when you attempted to work through that solution, where you got stuck, etc. Without that, your question is simply a duplicate. $\endgroup$
    – Lee Mosher
    Jan 31, 2021 at 15:14
  • $\begingroup$ Which I did in my question. Solution is clear. If you follow those steps you receive the answer, no magics. But what does the solution mean? How can he (runner) have exactly 17 runs in consecutive days (he runs daily-atleast 1 round)? What do I miss here? $\endgroup$
    – mig001
    Jan 31, 2021 at 15:51
  • $\begingroup$ I think I need some conceptual clarity :-( $\endgroup$
    – mig001
    Jan 31, 2021 at 16:04
  • 1
    $\begingroup$ To have exactly 17 runs in consecutive days is explained in that question and its solutions. Read in particular the last paragraph of the answer of André Nicolas. $\endgroup$
    – Lee Mosher
    Jan 31, 2021 at 16:40
  • $\begingroup$ I got it Thanks. $\endgroup$
    – mig001
    Feb 4, 2021 at 3:38


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