A possible chain of reasoning

Transcription

1 A possible chain of reasoning The ratio of the number of cuboids that touch the ground to the number that do not touch the ground is 1:2. Therefore the total number of cuboids is a multiple of 3. The total number of cuboids is less than 15. Therefore the total number of cuboids is 3 or 6 or 9 or 12. Seven cuboids are part of, or support, the centre Therefore the total number of cuboids is 9 or 12. The total number of cuboids is 9 or 12. Let s suppose that it is 9. Seven cuboids are part of, or support, the centre There are three platforms. The centre platform is the highest So only 2 cuboids are left to be part of, or support, the two lower platforms. So each of the lower platforms must consist of just one cuboid. But Each platform is at a different height above the ground. Therefore it cannot be true that each of the lower platforms consists of just one cuboid. Therefore the total number of cuboids is NOT 9. (proof by reductio ad absurdum) Therefore the total number of cuboids is 12. The ratio of the number of cuboids that touch the ground to the number that do not touch the ground is 1:2.

2 The total number of cuboids is 12. So by sharing the 12 cuboids into two groups in the ratio 1:2, we deduce that 4 cuboids touch the ground and 8 cuboids do not touch the ground. The width of each cuboid is one metre. The length of each cuboid equals its width. Therefore the length of each cuboid is one metre, and two opposite faces of the cuboid are squares. The ratio of the height of each cuboid to its length is 2:5. The length of each cuboid is one metre. Therefore the height of each cuboid is 40 cm = 0.4 m. The width of each cuboid is one metre. The length of each cuboid is one metre. Therefore each cuboid has two faces that are one metre squares, and four faces that are 1 m by 0.4m rectangles. The plan of the podium is a rectangle. All the cuboids fit together face to face. No cuboid overhangs any cuboid that supports it. 4 cuboids touch the ground. Each cuboid has two faces that are one metre squares, and four faces that are 1 m by 0.4m rectangles. Therefore the rectangular plan can be formed by four rectangles, some or all of which may be 1 m by 0.4m rectangles or one metre squares, placed edge to edge. The plan of the podium is a rectangle. The shortest edge of the plan of the podium is one tenth of its perimeter.

3 The only arrangement of four rectangles, some or all of which may be 1 m by 0.4m rectangles or one metre squares, placed edge to edge that forms a rectangle in which the shortest edge is one tenth of its perimeter is four metre squares placed in a row. (proof by exhaustion of possibilities) Therefore the 4 cuboids that touch the ground are fitted together in a row with their square faces horizontal. Cuboids are fitted together to make the podium. The cuboids are all the same shape. The cuboids are all the same size. The podium can be split into parts so that each part forms one No cuboid is part of, or supports, more than one There are three platforms. So we can try to work out how the identical cuboids are fitted together to support and form separately each of the three platforms. Two platforms are squares. No cuboid overhangs any cuboid that supports it. The 4 cuboids that touch the ground are fitted together in a row with their square faces horizontal. Therefore two of the platforms are the top faces of a single cuboid, which is the top cuboid of a pile of one or more cuboids placed one upon the other all with their square faces horizontal. The number of cuboids that are part of, or support, the silver medalist s platform is one third of the number forming the rest of the podium So the ratio of the number of cuboids that are part of, or support, the silver medalist s platform to the number of cuboids forming the rest of the podium is 1:3. Therefore, sharing 12 in the ratio 1:3 we deduce that 3 cuboids are part of, or support, the silver medalist s Either one or two of these is a cuboid that touches the ground. All the cuboids fit together face to face. Therefore the silver medalist s platform is the top of a pile of 3 cuboids placed one upon the other all with their square faces horizontal. (by visualising, sketching or making models of all possibilities) 3 cuboids are part of, or support, the silver medalist s

4 Seven cuboids are part of, or support, the centre The gold medallist stands on the centre Therefore 12 (7 +3) = 2 cuboids are part of, or support, the bronze medalists One of these is a cuboid that touches the ground. Therefore the bronze medalist s platform is the top of a pile of 2 cuboids placed one upon the other all with their square faces horizontal. (by visualising, sketching or making models of all possibilities) The bronze medalist s platform is the top of a pile of 2 cuboids placed one upon the other all with their square faces horizontal. Therefore the height of the bronze medallists platform is 0.8 m. The silver medalist s platform is the top of a pile of 3 cuboids placed one upon the other all with their square faces horizontal. Therefore the height of the silver medallists platform is 1.2 m. A cuboid s height is twice the difference between the heights of the gold and silver medalists platforms. Therefore the difference between the heights of the gold and silver medalists platforms is 0.2 m. The centre platform is the highest The gold medallist stands on the centre Therefore the height of the gold medallists platform is 1.2 m m = 1.4 m. Check 1 The 4 cuboids that touch the ground are fitted together in a row with their square faces horizontal. One of these is the bottom cuboid of the pile of 3 cuboids supporting and forming the silver medalists platform, and another is the bottom cuboid of the pile of 2 cuboids supporting and forming the bronze medalists Therefore the bottom of the structure supporting and forming the gold medalists platform consists of two cuboids placed together with their square faces horizontal.

5 The gold medallist stands on the centre Seven cuboids are part of, or support, the centre Therefore five cuboids are fitted together on top of these two base cuboids to form the centre All the platforms are flat Each cuboid has two faces that are one metre squares, and four faces that are 1 m by 0.4m rectangles. 5 x 0.4m = 2m Therefore the five cuboids can be fitted together with their square faces touching to form a cuboid with length 2m, width 1m and height 1m. That cuboid will fit on the two base cuboids to form the structure supporting the gold medalists (by visualising, sketching or making models of all possibilities) Therefore the height of the gold medalists platform is 0.4 m + 1 m = 1.4 m. Check 2 A cuboid s height is two thirds of the difference between the heights of the gold and bronze medalists platforms. Therefore the difference between the heights of the gold and bronze medalists platforms is one and a half times a cuboid s height. One and a half times 0.4 m = 0.6 m. The height of the bronze medallists platform is 0.8 m. Therefore the height of the gold medalists platform is 0.8 m m = 1.4 m