Olympiad Math Problem
This is a question from Arthur Engel: "Prove that an axb chessboard can be covered by nx1 tiles iff n|a or n|b". I have seen the solution but am not 'convinced' by it. Can anyone please help?
I@Apu
INote for OP
+solved @user1 @user2...
Πwhat solution did you see
Πthere is a paper on that problem, with 10-15 different solutions
S
SIt is something like this: Colouring the pieces of the chessboard as shown and then I tried making a system of equations to show some contradiction.
Πaahhh
Πgood proof
Πif n divides a or b then you can basically cut the board into pieces
that part is obvious to you right?
that part is obvious to you right?
SYes.
Πgreat
Πsee it this way
Πtake n different colors
Πstart painting the board, alternating between colors
Πif every color appears equal number of times, we are done
Πthat makes sense?
SYeah.
Πgreat
Πtry to see it this way now
Πif n does not divide a or b
let a= cn +r1
let a= cn +r1
Πnow you can basically prove that there will be some colors with unequal number of squares
Πthis is basically that proof
SOh.
SThanks a lot.
Πi hope this makes sense i’m not able to put it very well
Πlike if it is cn+r every color will be c times
Πexcept some
Πthats what the proof is
Πit’s very rigorous
Πug math required (not necessarily though 
)
)SThat is a good explanation. Thanks.
Πit’s not mine
Πi have worked on that book
Πi read a paper related to this long back
Πthe explanation in that was similar
i just put it like that
i just put it like that
Πwait let me cite
SYeah. I saw the solution in the book, but got lost somewhere in between. Now I see what the author was trying to say.
ΠPacking a Rectangle with Congruent N-ominoes DAVID A. KLARNER
McMas'ter University, Hamilton, Ontario, Canada Communicated b)' Gian-Carlo Rota Received April 1968
ABSTRACT
Golomb defines a polyomino (a generalized domino) to be a finite set of rook-wise connected cells in an infinite chessboard. In this note we discuss problems concerning the packing of rectangles with congruent polyominoes. Many of these problems are unsolved.
McMas'ter University, Hamilton, Ontario, Canada Communicated b)' Gian-Carlo Rota Received April 1968
ABSTRACT
Golomb defines a polyomino (a generalized domino) to be a finite set of rook-wise connected cells in an infinite chessboard. In this note we discuss problems concerning the packing of rectangles with congruent polyominoes. Many of these problems are unsolved.
Πthis is a very good proof
SWill have a look at this too. Once again, thanks a ton.
Πanytime :))
S+solved @πrate
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