Figure A 591014 a gives a square of 25 fields. This figure should be designed with (congruent) L-stones, whereby a special field is removed beforehand. An L-shaped stone consists of three square shaped L-shaped boxes (see illustration) A 591014 b).

a) One of the gray boxes is removed from the square. Show that the figure consisting of the remaining 24 squares can be laid with 8 L stones.

b) One of the white boxes is removed from the square. Show that the figure consisting of the remaining 24 squares can not be laid with 8 L stones.

I just cant solve this and cant find any solution on this question if you could help me, thanks so much and sorry if i have some grammer mistakes since im from Serbia and I'm 16 :)

• For part (a) you just have to trial-and-error. There are only 3 cases (removing a corner, a middle-of-side square, or the center) and each case is not difficult with just trial and error. Part (b) I'm not so sure. Usually these impossibility results involve a parity argument but nothing comes to my mind so far. However, I think it can be proven by (exhaustive) case analysis. Commented Sep 21, 2019 at 18:18
• Could you please writte me the steps for atleast a) so i can figure out how to do the same for b), if u have time Commented Sep 21, 2019 at 18:38

Big hint for (b)

Label the squares of the board like this:

A B A B A
C D C D C
A B A B A
C D C D C
A B A B A


Each $$\mathrm L$$-shaped stone tile covers at most one square labeled with an $$\mathrm A$$.

• +1. ha! how can i look at the original picture and still not find this "parity" argument? ;) Commented Sep 21, 2019 at 21:30
• Ye i tried with it so it means it has to contain atleast A and at some point i understand that it cant be laid but i still cant prove it the right way :/ Commented Sep 21, 2019 at 22:08

HINT

(a) is basically trial & error. here is a sample, the case when the removed gray square is at a corner. the three positions with the same number = an L-shaped piece. you just have to try the others 2 cases on your own. it's not that hard.

x 1 1 8 8
2 2 1 8 7
2 4 4 7 7
3 4 5 6 6
3 3 5 5 6

• Thanks, I made part a) done but what about b) since its now white box Commented Sep 21, 2019 at 20:22