greetings. since there has some been some discussion recently about the mrs. perkins quilt problem, i thought i would share some other open (as far as i know) problems involving tiling integer squares in squares. if you can add to my knowledge on any of these, it would be greatly appreciated. ------------------------------------------------------------------- 1) mrs. perkins quilt problem: pack a square of side n with the fewest number f(n) of co-prime squares. recently lance gay has shown f(n) <= 19 for 101 <= n <= 108. it is fairly easy to show that f(n) <= 20 for 109 <= n <= 132. this is problem C3 in UPIG. ------------------------------------------------------------------- 2) pack a square of side n with squares so that the smallest square is as large as possible. some partial results are at http://www.stetson.edu/~efriedma/mathmagic/1298.html. the values listed there are known to be best, by computer search. can anyone extend these values? it appears that the smallest square grows like n^(1/2). i had a custom wood puzzle made of a square of side 103 cut into smaller squares all of which are 15 and larger. ------------------------------------------------------------------- 3) can every sufficiently large square be cut into squares with no size square being used more than twice? it appears true for squares of size 22 and more, but seems difficult to prove. perhaps it is even true that all sufficiently large squares admit perfect tilings, so that only one square of each size is needed? ------------------------------------------------------------------- 4) partridge packings: can 1 square of side 1, 2 squares of side 2, up to n squares of side n, be packed into a square of side n(n+1)/2 ? it is not possible for 2 <= n <= 7, and it is possible for 8 <= n <= 34. is it possible for all larger n? ------------------------------------------------------------------- 5) what is the smallest value of n for which n squares each of sides 1-24 can pack n squares of side 70? it is well known that n=1 doesn't work. it must be easy to prove that this is possible for large n. what is the smallest value? i managed to pack 4 copies of 1-24 inside a square of side 140. ------------------------------------------------------------------- 6) diverse packings: what squares can be packed with 1 or 2 squares each of sides 1-n, for some n? some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/0602.html. it is conjectured that this is possible for n >= 34, but existence of infinitely many packings of this sort is still in doubt. do packings of this sort exist for every n >= 9? ------------------------------------------------------------------- 7) square strip packings: which squares can be packed with "strips" of squares of width 1-n, for some n? some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/1201.html. (a strip is a horizontally or vertically adjacent union of squares of a given size, so a 3x12 rectangle is a strip of width 3.) it seems clear that all squares of size 6 or more can be packed in this way, but can anyone prove this? ------------------------------------------------------------------- 8) decending squares: which squares can be packed with squares so that every square directly above a square is strictly smaller? the answer seems to be all except 2,4,5,7,8,9,11,13,15,17,19,21,23,29,33. is this correct? can anyone prove it? some of these make fabulous puzzles. ------------------------------------------------------------------- 9) anti-partridge packings: can n squares of side 1, (n-1) squares of side 2, up to 1 square of side n be packed inside a square without overlap? packings of n=6 and n=25 are given at: http://www.stetson.edu/~efriedma/anti/. this is only possible for certain values of n due to area considerations. the next possible case is n=96. is it possible? ------------------------------------------------------------------- since i didn't quite make a dozen, here are a few packing (rather than tiling) problems. ------------------------------------------------------------------- 10) can you prove the smallest square that can contain 12 non-overlapping unit squares is side 4? it is almost certainly true, and has been proved for 14 unit squares. this is part of problem D4 in UPIG. ------------------------------------------------------------------- 11) given integers n and m, find n squares of one size and m squares of another (possibly equal) size so that these n+m non-overlapping squares fit inside a unit square and have the largest possible area. some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/0400.html. some of the best known results involve tilted squares. surely some of these can be improved. ------------------------------------------------------------------- 12) what is the smallest square f(n) that contains squares of sides 1-n? does anyone even have a list of the best known values for f(n)? this is part of problem D5 in UPIG. ------------------------------------------------------------------- erich friedman