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Non-arithmetic Triplets (Posted on 2013-04-08) Difficulty: 3 of 5
Find nine different integers from 1 to 20 inclusive such that no combination of any three of the nine integers form an arithmetic sequence.

(For example, if two of the integers chosen were 7 and 13, then that would preclude 1, 10 and 19 from being included.)

No Solution Yet Submitted by K Sengupta    
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Hints/Tips re(3): Counting Up,,,,Why not up and down? | Comment 6 of 9 |
(In reply to re(2): Counting Up by Charlie)

My way:

I have solved it rather quickly, counting from both ends, using the following logic:

a.Clearly both 1 and 20 must be in the solution, otherwise the puzzle would address a shorter range of numbers.

b. Start coverging from both direction toward the middle:-
chose delta=1 on one end (say the begining) and=2 on the other.

1 2.....18 20    (later by replacing each member m by 21-m we will have another valid solution)

Now my task (so would be the computer task - but who cares?)became significantly easier - the (1,2 )delta set can be applied only in one way , got a dead end  - tried (2,2) and (1,3)etc
and after short fiddling with (4,3)==>  1  2  6  ...15 18 20
got the solution    1  2  6  7  9 14 15 18 20.

 Stopped here,
realising that there is at least one more symmetrical solution.

 

Edited on April 9, 2013, 1:24 am
  Posted by Ady TZIDON on 2013-04-09 01:20:20

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