$begingroup$

a) Is it doable to put the integers 1 to 25 in a 5 x 5 grid in order that no column or row incorporates an growing or reducing 3-term arithmetic development (A.P.)?

b) Can this be performed in a 6 x 6 grid with integers 1 to 36?

Note: Rows or columns can include numbers which belong to an A.P., however not themselves be in a monotone arithmetic development, i.e. 13 25 17 15 23 could be a row, however not 25 13 15 23 17.

$endgroup$

5

$begingroup$

@Prim3numbah has discovered a 5×5 grid. See their resolution right here

Below is the 6×6 grid that I’ve discovered

+--+--+--+--+--+--+
|11|06|09|08|15|20|
+--+--+--+--+--+--+
|05|28|26|27|34|25|
+--+--+--+--+--+--+
|07|29|01|02|36|23|
+--+--+--+--+--+--+
|16|27|03|04|35|24|
+--+--+--+--+--+--+
|13|32|30|31|33|22|
+--+--+--+--+--+--+
|12|10|17|18|21|14|
+--+--+--+--+--+--+

My technique was to

Place very excessive and really low numbers close to the center of the sq., after which place the center numbers across the edge. Since I used to be capable of efficiently implement this technique on my first try, I really feel assured that there are a number of different preparations that might work, though I’ve no proof of this.

Note that

I did my greatest to double-check my reply, however I could have missed a development, since I used to be attempting to do that with out a pc. Please drop a remark in case you discover an error.

$endgroup$

3