Article 19167 of sci.math:
Date: Wed, 16 Dec 1992 22:40:42 GMT

[Draw integers randomly from 1 to 100. We want to get a pair differing by 3.]

Several people have already observed that 51 draws is insufficent to
guarantee a pair three apart.

e.g.   1 2 3, 7 8 9, 13 14 15, .....   91 92 93,  97 98 99  might be drawn.

It has not actually been proved that 52 draws is certain to give the desired
result.

Here is the proof. (Pigeonhole principle).
~~~~~~~~~~~~~~~~~
Partition the integers into 51 subsets:

{1,4}   {2,5}   {3,6}
{7,10}  {8,11}  {9,12}
.....
{91,94} {92,95} {93,96}
{97,100}
{98}
{99}

Now when 52 numbers are drawn, at least 2 must come from one subset, and thus
be 3 apart.

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Bill Taylor.       wft@math.canterbury.ac.nz 
Bill Trylor. que  rwft@maih.casterkury.aa.n! 
Tiel Tryloco quer rwst@maihuc sterkery.ga.n!
Thelworyd co quer rwsi@mvihus strikesy.gain!
The world conqueror sig-virus strikes again!
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