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wu :: forums    riddles    putnam exam (pure math) (Moderators: towr, SMQ, Grimbal, Eigenray, Icarus, william wu)    GMO 2006 question « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: GMO 2006 question  (Read 580 times) birbal Full Member     Gender: Posts: 250 GMO 2006 question   « on: Dec 4th, 2006, 8:21pm » Quote Modify This is a question from GMO 2006.   Given a square grid of size 6 x 6. The grid is divided into 9 rectangles with integer size lengths. Prove that at least two rectangles are congruent to each other. IP Logged The only thing we have to fear is fear itself! Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: GMO 2006 question   « Reply #1 on: Dec 5th, 2006, 6:49am » Quote Modify Let's see... What are the nine smallest rectangles we could possibly have? To how many unit square do they add up to? IP Logged birbal Full Member     Gender: Posts: 250 Re: GMO 2006 question   « Reply #2 on: Dec 5th, 2006, 8:36pm » Quote Modify Ok... the nine smallest incongruent rectangles can be 1x1, 1x2, 1x3, 1x4, 1x5, 1x6, 2x2, 2x3, 2x4   and the total square unit they add up to are   39 units. mm but the total area is 36 units. Cool!! So this is not possible??   I am not sure if this is proved. IP Logged The only thing we have to fear is fear itself! SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: GMO 2006 question   « Reply #3 on: Dec 6th, 2006, 6:02am » Quote Modify More formally: 1) Assume that no two rectangles are congruent. 2) The nine smallest incongruent rectangles by area are 1x1, 1x2, 1x3, 1x4, 2x2, 1x5, 1x6, 2x3, and 1x7. 3) These rectangles have a total area of 38 square units. 4) Therefore no set of nine incongruent rectangles can tile a 6x6 square, and premise (1) must be false. 5) Therefore its negation must be true: at least two rectangles are congruent. Q.E.D.   (And if we consider 1x1 and 2x2 to be congruent, the situation only gets worse as the smallest rectangles are then 1x1, 1x2, 1x3, 1x4, 1x5, 1x6, 2x3, 1x7, and 1x8 for a total area of 40.)   --SMQ IP Logged --SMQ towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: GMO 2006 question   « Reply #4 on: Dec 6th, 2006, 6:35am » Quote Modify on Dec 6th, 2006, 6:02am, SMQ wrote: 2) The nine smallest incongruent rectangles by area are 1x1, 1x2, 1x3, 1x4, 2x2, 1x5, 1x6, 2x3, and 1x7. 1x7 doesn't fit in a 6x6 square, so picking 2x4 makes more sense. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: GMO 2006 question   « Reply #5 on: Dec 6th, 2006, 8:37am » Quote Modify Well, that just means he proved the generalization where you're allowed to rearrange the pieces into rectangles. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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