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wu :: forums    riddles    easy (Moderators: Icarus, Grimbal, SMQ, towr, ThudnBlunder, william wu, Eigenray)    An Onerous Fence « Previous topic | Next topic » Pages: 1 2  Reply Notify of replies Send Topic Print    Author  Topic: An Onerous Fence  (Read 50511 times) SWF Uberpuzzler     Posts: 879 Re: An Onerous Fence   « Reply #25 on: Nov 7th, 2016, 5:28pm » Quote Modify How about (333...)*(333...) = 111...   There are no final digits, thus avoiding the mod 4 issue.  I guess that doesn't count.   IP Logged rloginunix Uberpuzzler     Posts: 1029 Re: An Onerous Fence   « Reply #26 on: Nov 9th, 2016, 6:45pm » Quote Modify I think I got it but let us see if we can destroy this.   Lemma.Let en be an arbitrary even number > 2. Then: en2 = 0, 4 (mod 16) (direct) Proof. A given even number is a product of 2 and either an even number 2q or an odd number 2q+1 where q = 1, 2, 3, ...   In the first case: ene = 2*2q = 4q, q = 1, 2, 3, ... ene2 = 16q2 = 0 (mod 16) In the second case: eno = 2*(2q + 1), q = 1, 2, 3, ... eno2 = 4(4q2 + 4q + 1) = 16q2 + 16q + 4 = 16(q2 + q) + 4 = 4 (mod 16) What was required to prove.   For all fours, proof by contradiction. By hypothesis, a number composed of nothing but all fours is an even square, the number 44 being the smallest one of interest: 44 = 40 + 4 = 16*2 + 8 + 4 = 16*2 + 12 = 12 (mod 16) 444 = 400 + 44 = 4*100 + 44 = 4*4*25 + 44 = 16*25 + 16*2 + 12 = 12 (mod 16) 4444 = 4000 + 400 + 44 = 16*250 + 16*25 + 16*2 + 12 = 12 (mod 16) 44444 = 40000 + ... = 12 (mod 16) etc. Contradiction: 12 is neither 0 nor 4. Hence, a number composed of nothing but all fours, 44 being the smallest such number, can not be a perfect square.   (shoot) IP Logged rloginunix Uberpuzzler     Posts: 1029 Re: An Onerous Fence   « Reply #27 on: Nov 12th, 2016, 9:42am » Quote Modify Then: 122 = 144 382 = 1444 Can this pattern be continued? IP Logged dudiobugtron Uberpuzzler     Posts: 735 Re: An Onerous Fence   « Reply #28 on: Nov 12th, 2016, 11:25am » Quote Modify on Nov 12th, 2016, 9:42am, rloginunix wrote: Then: 122 = 144 382 = 1444 Can this pattern be continued? My calculator says that the square root of 14444 is not an integer. IP Logged rloginunix Uberpuzzler     Posts: 1029 Re: An Onerous Fence   « Reply #29 on: Nov 12th, 2016, 1:01pm » Quote Modify I was hoping for some deductions,   and then something like:   ... the longest string of fours that can terminate a square ... « Last Edit: Nov 15th, 2016, 7:03am by rloginunix » IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: An Onerous Fence   « Reply #30 on: Nov 12th, 2016, 1:23pm » Quote Modify So like what I said in reply #10? « Last Edit: Nov 12th, 2016, 1:24pm by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB rloginunix Uberpuzzler     Posts: 1029 Re: An Onerous Fence   « Reply #31 on: Nov 12th, 2016, 1:46pm » Quote Modify Dangit ... the tread's getting too long (and I am getting too old). IP Logged Pages: 1 2  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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