Chủ đề này có 2 trả lời

[bluesea]

solution from Christian Sattler(Germany): first we define an involution on the hooks: send every hook to the one which includes the square where the hook goes around. it is clear that this is an involution, so the number of hooks is even. it immediately follows that 12 | mn. there are three cases(wlog 3 divides m): 3 divides m and 4 divides n(trivial since we can do 3x4-rectangles), 12 divides m, then n can be arbritarily, but not equal to 1,2 or 5(it is clear that n is not 1,2, and 5 is easy to exclude. in all other cases we may write n as linear combination of 3 and 4 with positive coefficients and we can use 3x4-rectangles again taking rows of 3x4-rectangles or 4x3-rectangles). so we are left with the case 6 | m and 2 | n. so look again at our involution: it shows that our hooks appear in pairs of the possible forms: either 3x4-rectangles or something similar to an S:
.===
.===
===.
===.
now mark every second row. it is easy to see that there are always 6 out of the 12 squares of the S are marked, the same for horizontal 3x4-rectangles, but vertical 3x4-rectangles have either 4 or 8 squares marked, that shows that there's an even number of them. doing the same with columns, we get that the total number of 3x4-rectangles is even. now, look at the coloring of
.==.
=..=
.==.
=..=
and periodic. it is easy to see that 6 out of 12 squares are marked at 3x4-rectangles, the S upright, the S turned by 90 degree and even distance to the left boundary of the rectangle, but 4 or 8 for an S turned by 90 degree with odd distance to the left boundary. this shows that there's an even number of them and by translating the coloring by 1 to the right and turning it all around, we get that the number of S is even as well. this shows that 24 is a divisor of mn, so we get one of the two other cases


Bạn nào dịch hộ tớ với,cám ơn

[FDF]

Bài này chỉ khó ở chỗ tìm cách tô màu để chứng minh 6a*2b lát được thì a hoặc b chẵn.Lời giải trên đưa ra hai cách tô:
1)tô mọi hàng ở vị trí chẵn->số hcn 3*4 chẫn
2)tô(2i+1;3j+1) và (2i;3j+1);(2i;3j+2)-> số hình S chẵn

Bài viết đã được chỉnh sửa nội dung bởi FDF: 10-02-2006 - 10:19

[dhkhtn-tnt]

Cái này là do Peter Scholze post trên ML,mình có chép về n0 ko hiểu lắm!!!