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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).

enter image description here

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 :)

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  • $\begingroup$ 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. $\endgroup$
    – antkam
    Commented Sep 21, 2019 at 18:18
  • $\begingroup$ 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 $\endgroup$ Commented Sep 21, 2019 at 18:38

2 Answers 2

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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 $\mathrm A$.

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  • $\begingroup$ +1. ha! how can i look at the original picture and still not find this "parity" argument? ;) $\endgroup$
    – antkam
    Commented Sep 21, 2019 at 21:30
  • $\begingroup$ 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 :/ $\endgroup$ Commented Sep 21, 2019 at 22:08
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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
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  • $\begingroup$ Thanks, I made part a) done but what about b) since its now white box $\endgroup$ Commented Sep 21, 2019 at 20:22

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