View topic - Again Warm-Up Prob for a TST-Partition of a set into triples • Art of Problem Solving

(a) suppose that if $D_{n}=\{1,2,...,n\}$ has property $D$ , then so have $D_{4n}$ and $D_{4n+3}$ :

suppose $D_{n}$ has property $D$ .
$D_{4n}$ can be partitioned into the following sets (in which the last integer is the sum of the two first :
$\{1,2n+n,2n+n+1\}$
$\{3,2n+n-1,2n+n+2\}$
$\{5,2n+n-2,2n+n+3\}$
...
$\{2n-1,2n+1,4n\}$
and eventually the even integers : $\{2,4,6,...,2n\}=2D_{n}$
if $D_{n}$ has property $D$ , $2D_{n}$ also has property $D$ and hence $D_{4n}$ has property $D$

$D_{4n+3}$ can be partitioned in a similar way :
$\{1,2n+n+2,2n+n+3\}$
$\{3,2n+n+1,2n+n+4\}$
$\{5,2n+n,2n+n+5\}$
...
$\{2n+1,2n+2,4n+3\}$
and the even integers : $\{2,4,6,...,2n\}=2D_{n}$

so we have proved that if $D_{n}$ has property $D$ , so have $D_{4n}$ and $D_{4n+3}$
now it is sufficient to check that :
- $D_{3}$ has property $D$
- 3x4=12
- 12x4+3=51
- 51x4+3=207
- 207x4+3=831
- 831x4=3324
From that we conclude $D_{3324}$ has property $D$ .

(b) It is clear that $D_{n}$ cannot have property $D$ if the sum of its elements is odd : suppose $D_{n}$ has property $D$ ; by construction, the sum of the elements of each triple in the partition is even so the same must be true of $D_{n}$ . So $D_{3309}$ cannot have property $D$ because the sum of its elements is an odd integer.

Again Warm-Up Prob for a TST-Partition of a set into triples

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