tag:blogger.com,1999:blog-8231784566931768362.post7207057826920903778..comments2023-09-09T08:21:55.454-04:00Comments on MathNotations: A 'Simple' Traversal through a Number Grid -- Patterns, Functions, Algebra Investigation Part IDave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-8231784566931768362.post-29237582412566089132008-03-28T01:28:00.000-04:002008-03-28T01:28:00.000-04:00Thanks for the vocab word, Eric. If I get a chanc...Thanks for the vocab word, Eric. If I get a chance to do this with my middle schoolers, I'll definitely get their English teacher to use this as their word of the day too. :)mathmomhttps://www.blogger.com/profile/05869925405540832241noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-21766498941776826572008-03-27T06:38:00.000-04:002008-03-27T06:38:00.000-04:00Dave, T(n) is the value of the cell we visit after...Dave, <BR/><BR/>T(n) is the value of the cell we <BR/>visit after the n-th step. And<BR/>n is simply a value that is in<BR/>the table and of which we can <BR/>compute the position.<BR/><BR/>The idea is to find the position of <BR/>n and if it lies in a even row T(n)<BR/>is different than n (because we go <BR/>from right to left in even rows).<BR/><BR/>I hope that explains it better.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-22894250069475952722008-03-27T01:37:00.000-04:002008-03-27T01:37:00.000-04:00On the subject of Boustrophedon, check out www.re...On the subject of Boustrophedon, check out<BR/> www.research.att.com/~njas/doc/bous.pdf<BR/><BR/>or read the wikipedia page on Boustrophedon transform.<BR/><BR/>(I'm a big fan of Neil Sloane and the OEIS)Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-60111835382977059032008-03-27T00:28:00.000-04:002008-03-27T00:28:00.000-04:00Eric, Boustrophedon has to be my new favorite word...Eric, <BR/>Boustrophedon has to be my new favorite word - even better than quomodocumque! I may actually use this left-right right-left text method in my next post. Now if we also write palindromically, we'll really confuse everyone!<BR/>A MAN A PLAN A CANAL PANAMADave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-42182122622630941312008-03-27T00:15:00.000-04:002008-03-27T00:15:00.000-04:00Nice analysis, Florian! I appreciate the effort an...Nice analysis, Florian! I appreciate the effort and thinking that went into it as well as the logic and clarity of your explanation. <BR/><BR/>Your formula looks strong but the meaning of 'n' may be getting ambiguous. For example, when a=7, n=8, you computed x=1 and y=2, which corresponds to the 1st column of the 2nd row. Yes, the number 8 does appear there but, in my notation, n = 8 refers to the 8th cell visited in the matrix which ends up being in the 7th column of the 2nd row. The value of that cell is of course 14. I think I'm getting confused by your interpretation of n. Clearly your method works.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-49031248913260643642008-03-26T23:34:00.000-04:002008-03-26T23:34:00.000-04:00You should introduce these students to the new wor...You should introduce these students to the new word of the day: boustrophedon.<BR/><BR/>adjective & adverb<BR/>(of written words) from right to left and from left to right in alternate lines.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-34029327531315613432008-03-26T22:25:00.000-04:002008-03-26T22:25:00.000-04:00This is very interesting!For a table with width a ...This is very interesting!<BR/><BR/>For a table with width a and n=0a+1,2a+1,4a+1, ...<BR/>we get the following mapping for 1,2,3,...to T(n):<BR/><BR/>(going from left to right in the table)<BR/> n |-> n<BR/> ...<BR/> n+ a-1 |-> n+ a-1<BR/>(going from right to left in the table)<BR/> n+ a |-> n+2a-1<BR/> ...<BR/> n+2a-1 |-> n+ a<BR/><BR/>This isn't really handy for calculating T(n). But<BR/>if we knew the (x,y)-Position of each value n<BR/>in the table we could write T(n) as a formula.<BR/><BR/>Look sharp and see that for each value n=1,2,...<BR/>the x and y positions are:<BR/><BR/> x = (n-1) mod a +1 (with 1 =< x =< a)<BR/> y = floor((n-1)/a) +1 (with 1 =< y)<BR/><BR/>And that n can now be expressed as:<BR/><BR/> n = y*a + x.<BR/><BR/>If y is odd then T(n) = n (this is a row where we go from left to right)<BR/>and if y is even T(n) = y*a + (1-x) (this a row where we go from right to left)<BR/><BR/>Examples:<BR/><BR/><BR/>1) a=2, n=100:<BR/><BR/> Step 1: Get (x,y)-Position of n:<BR/><BR/> x = 99 mod 2 +1 = 2<BR/> y = floor(99/2)+1 = 50<BR/><BR/> (This means that n is in the 50th row and 2nd column of the table.)<BR/><BR/> Step 2: Determine how to calculate T(n)<BR/><BR/> y is even so T(n) = y*a + (1-x)<BR/><BR/> Setp 3: Calculate T(n):<BR/><BR/> T(100) = 50*2+(1-2) = 99<BR/><BR/>2) a=2, n=153:<BR/><BR/> Step 1: Get (x,y)-Position of n:<BR/><BR/> x = 152 mod 2 +1 = 1<BR/> y = floor(152/2) +1= 77<BR/><BR/> (This means that n is in the 77th row and 1st column of the table.)<BR/><BR/> Step 2: Determine how to calculate T(n)<BR/><BR/> y is odd so T(n) = n.<BR/><BR/> Setp 3: Calculate T(n):<BR/><BR/> T(153) = 153.<BR/><BR/>3) a=2, n=999:<BR/><BR/> Step 1: x = 998 mod 2 +1 = 1<BR/> y = floor(998/2) +1 = 500<BR/> Step 2: y is even so T(n) = y*a + (1-x)<BR/> Step 3: T(999) = 500*2 +(1-1) = 1000.<BR/><BR/>4) The formula works for any integer a > 0. <BR/> a = 7, n=8: (By drawing we expect T(8)=14)<BR/><BR/> Step 1: x = 7 mod 7 +1 = 1<BR/> y = floor(7/7)+1 = 2<BR/> Step 2: y is even so T(n) = y*a + (1-x)<BR/> Step 3: T(8) = 2*7 +(1-1) = 14.<BR/> Which is the expected result.<BR/> <BR/>I think a proof is not necessary here, the formulas<BR/>are very straight forward once you see how they work<BR/>with the indexes in the rows that read from right to left.Anonymousnoreply@blogger.com