tag:blogger.com,1999:blog-15628310.post115746259381663502..comments2021-04-15T02:41:46.574-04:00Comments on Question of the day: SquaresAnonymoushttp://www.blogger.com/profile/18153935609499338685noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-15628310.post-64836581621277707732009-05-18T02:24:00.000-04:002009-05-18T02:24:00.000-04:00Jonathan you forgot 3Jonathan you forgot 3Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15628310.post-1157841658278747342006-09-09T18:40:00.000-04:002006-09-09T18:40:00.000-04:00Since you told us there are 37 duds, lets find the...Since you told us there are 37 duds, lets find them. The 37th will be it.<BR/><BR/>2, 6, 7, 8, 11, 12, 15, 18, 19, 22<BR/>(that's ten)<BR/>23, 24, 27, 28, 31, 32, 33, 40, 43,<BR/><BR/>[Digression: is 44 on the list? 44 = 36 + 8, but 8 is on the list. 44 = 25 + 19, but 19 is on the list. 44 = 16 + 28....]<BR/><BR/>44<BR/>(that's twenty)<BR/>47, 48, 51, 55, 60, 67, 72, 76, 88, 92, <BR/><BR/>(that's thirty)<BR/>96, 108, 112, 115, 124, 128.<BR/><BR/>Where'd I goof? I only got 36 up to 128.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-15628310.post-1157637051411731872006-09-07T09:50:00.000-04:002006-09-07T09:50:00.000-04:00Abe, I'll have to think about that some more.Abe, I'll have to think about that some more.Anonymoushttps://www.blogger.com/profile/18153935609499338685noreply@blogger.comtag:blogger.com,1999:blog-15628310.post-1157568821346479092006-09-06T14:53:00.000-04:002006-09-06T14:53:00.000-04:00I was thinking of a mathematical way to solve this...I was thinking of a mathematical way to solve this, since every square can be expressed as a sum of consecutive odd numbers:<BR/><BR/>1^2=1<BR/>2^2=4=1+3<BR/>3^2=9=1+3+5<BR/>4^2=16=1+3+5+7<BR/>etc.<BR/><BR/>since (n+1)^2=n^2+2n+1, meaning that (n+1)^2 is 2n+1 greater than n^2, which is where the consecutive odd numbers come from. Therefore, every square can be expressed like the sum from n=0 to some number k of (2n+1). Anyway, there has got to be some way to prove this like that, but I didn't have time yesterday. Have you tried that, Mike? Any thoughts?Abehttps://www.blogger.com/profile/04424868492071587450noreply@blogger.comtag:blogger.com,1999:blog-15628310.post-1157545101211466582006-09-06T08:18:00.000-04:002006-09-06T08:18:00.000-04:00Ohh... No answer? Did I finally stump you all?128...Ohh... No answer? Did I finally stump you all?<BR/><BR/>128 is the largest number: Note that 128-100 = 28 which cannot be expressed. But 29 - 50 can be (for instance 29 = 25 + 4, 30 = 25 + 4 + 1, etc...)Anonymoushttps://www.blogger.com/profile/18153935609499338685noreply@blogger.com