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You must place 7 robot ants on a long rod and set each of them to move left or right starting at time 0. You can set any positive speed for each ant. When 2 or more ants meet, they turn around. When an ant reaches the end it falls off.

Show that you can place and set the ants in such a way that none of them ever falls off.


Bonus question: Can you do this for other odd numbers, too?

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7
  • 1
    $\begingroup$ Since someone tried to wiggle out of the intended wording with the last problem, perhaps clarify that "left or right" is along the direction of the rod/parallel to its run/equivalent to moving along a long but not infinite axis $\endgroup$
    – bobble
    May 29, 2022 at 4:40
  • $\begingroup$ What happens if I put two ants with the same starting position and direction and speed? $\endgroup$
    – justhalf
    May 29, 2022 at 4:56
  • $\begingroup$ @justhalf Starting positions must be different. $\endgroup$
    – Eric
    May 29, 2022 at 5:14
  • $\begingroup$ I have found that for 3 ants is impossible. Interesting problem. $\endgroup$
    – justhalf
    May 29, 2022 at 5:32
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    $\begingroup$ @justhalf how? can you elaborate a bit $\endgroup$
    – I'm Nobody
    May 29, 2022 at 6:56

1 Answer 1

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Picture showing how to place 4,7 or 10 ants:

enter image description here

x-axis is time, y-axis is position. The 4 black ants alone are viable as are the blacks together with the 3 blues or all 10.

Ants here move either at speed 1 or 1/3. Except for the first and last ones, the slower ants can be left out.

Therefore this construction yields solutions for 4, 6, 7, 8, 9, 10, 11, ... ants.

P.S.: 5 is also possible:

enter image description here

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5
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    $\begingroup$ One of those where a well presented diagram says it all. Great answer! $\endgroup$
    – hexomino
    May 30, 2022 at 21:07
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    $\begingroup$ Worth noting that this involves instances of three ants meeting. (Yes, this is allowed by the rules.) (Maybe 3+ ants meeting is needed to get 7 or any odd number?) $\endgroup$
    – tehtmi
    May 30, 2022 at 23:31
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    $\begingroup$ Does this mean that 5 ants is not possible? $\endgroup$ May 31, 2022 at 1:20
  • $\begingroup$ @DmitryKamenetsky No, it only gives positive answers. $\endgroup$ May 31, 2022 at 7:29
  • $\begingroup$ Congratulations! You've solved the most difficult 5 ants case! $\endgroup$
    – Eric
    May 31, 2022 at 14:28

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