<s>[INST] The number of ways to put numbers in squares If  one put the numbers from $1$ to $7$ in the squares in stead of letters in which the sum of cells is in descending order like shown in the figure .Every number should  be used once or twice or never .Also the order of numbers in every oblique row its sum appear always in descending order.
In how many way one can do that ?

 [/INST] If the inequalities are strict, $Cxy$ can only be $431$ or $521$  Starting with $521$, we must have $b=6, A=7$ and $AbCD$ fails.  So we must have $Cxy=431$.  Then $cxz=432$ or $531$  If it is $531$ we must have $abcd=7651$, but then $A=7$ and again $AbCD$ fails.  If $b=6, A=7$ and again $AbCD$ fails, so $b=5$.  Then $a=7, d=3$ and we can have either $A=7,D=1$ or $A=6, D=2$, giving two solutions.  $$\begin {array}{c c c} A & 7 & 6 \\ a & 7& 7\\b & 5 & 5 \\c&4&4\\C&4&4\\d&3&3\\D&1&2\\x&3&3\\y&1&1\\z&2&2\end {array}$$
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