---------- Forwarded message ---------- From: geoffrey morley <ghmorley@gmail.com> Date: Sat, Feb 18, 2012 at 5:29 PM Subject: How much water can a 'cubed square' hold? To: Stuart Anderson <stuart.errol.anderson@gmail.com> Stuart, Place a (solid) cube on each subsquare of the perfect squared squares of order 22, so that each subsquare is the base of a cube. http://www.squaring.net/sq/ss/spss/o22/spsso22.html Each structure now comprises 22 cubes. Fill each structure with as much water as it can hold. How many ponds does each structure have? Which structure's ponds hold the most water? See also http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces 22:110A(60,50)(23,27)(24,22,14)(7,16)(8,6)(12,15)(13)(2,28) (26)(4,21,3)(18)(17) 3 ponds, volume of water is 540+2404+80=3024. z=height of water level in pond above bases of cubes=15, covers cubes (3,12), v=volume of water=(15-3)*3^2+(15-12)*12^2=540; z=16, cubes (4,6,7,8,13,14), v=2404; z=22, cube (2), v=80. 22:110B(60,50)(27,23)(24,22,14)(4,19)(8,6)(3,12,16)(9)(2,28) (26)(21)(1,18)(17) 2 ponds, volume of water is 3534+80=3614. z=17, cubes (1,3,4,6,8,9,12,14,16), v=3534; z=22, cube (2), v=80. 22:139A(80,59)(21,38)(29,28,17,27)(7,10)(18,20)(4,3)(32,8) (1,31)(30)(24,2)(22) 3 ponds, volume of water is 1488+8800+27=10315. z=20, cubes (2,8,18), v=1488. z=27, cubes (3,4,7,10,17,21), v=8800. z=28, cube (1), v=27. 22:147A(55,44,48)(40,4)(52)(26,29)(23,3)(20,31,21)(5,47)(43) (9,17)(1,8)(32)(25) 3 ponds, volume of water is 1380+8204+576=10160. z=23, cubes (3,20), v=1380. z=25, cubes (1,5,8,9,17,20,21), v=8204. z=40, cube (4), v=576. 22:147B(59,43,45)(41,2)(47)(34,25)(21,37,8)(55)(22,12)(10,23) (32)(11,26)(19,4)(15) 4 ponds, volume of water is 660+3081+1856+156=5753. z=15, cubes (4,11), v=660. z=22, cubes (10,12,21), v=3081. z=37, cube (8), v=1856. z=41, cube (2), v=156. 22:154A(61,52,41)(11,30)(9,35,19)(46,24)(16,33)(22,2)(36,17) (50)(47,21)(5,31)(26) 2 ponds, volume of water is 6766+9640=16406. z=26, cubes (2,9,21,22,24), v=6766. z=30, cubes (11,16,17), v=9640. 22:172A(97,75)(22,53)(39,42,38)(9,44)(4,19,13,2)(36,3)(11) (33,16)(24)(1,18)(17) 4 ponds, volume of water is 480+2860+2700+7744=13784. z=17, cubes (1,4,16), v=480. z=19, cubes (2,9,11,13), v=2860. z=33, cube (3), v=2700. z=38, cube (22), v=7744. 22:192A(86,49,57)(41,8)(28,37)(19,9)(47,35,4)(31,14)(10,36) (17,26)(12,71)(62)(59) 1 pond, volume of water is 45549. z=36, cubes (4,8,9,10,12,14,17,19,26,28,31,35), v=45549. If the squares that are squared are all the same size (side = 1 unit, say), then the volume of water is 22:110A- 3024/110^3 = 0.002272 (3 ponds); 22:110B- 3614/110^3 = 0.002715 (2 ponds); 22:139A- 10315/139^3 = 0.003841 (3 ponds); 22:147A- 10160/147^3 = 0.003198 (3 ponds); 22:147B- 5753/147^3 = 0.001811 (4 ponds); 22:154A- 16406/154^3 = 0.004492 (2 ponds); 22:172A- 13784/172^3 = 0.002709 (4 ponds); 22:192A- 45549/192^3 = 0.006435 (1 pond). 22:192A has only one pond, but holds the most water! Obviously (think about it), no two ponds can have the same water level. Geoff Some obvious questions; Which 'cubed' squared square holds most water? ( This would be a candidate, http://www.squaring.net/gfx/5468o55TTF.png - found in 1940 by Tutte, Smith, Stone & Brooks) Which (normalised) 'cubed' squared square holds the record?