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
platform.
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
platform.
There are three platforms.
The centre platform is the highest platform.
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.
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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.
The height of each cuboid is 0.4 m.
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.
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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 platform.
No cuboid is part of, or supports, more than one
platform.
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
platform.
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 platform.
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Seven cuboids are part of, or support, the
centre platform.
The gold medallist stands on the centre
platform.
Therefore 12 – (7 +3) = 2 cuboids are part of, or support, the bronze medalists’ platform.
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.
The height of each cuboid is 0.4 m.
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.
The height of each cuboid is 0.4 m.
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.
The height of each cuboid is 0.4 m.
Therefore the difference between the heights of the gold and silver medalists’ platforms is 0.2 m.
The centre platform is the highest platform.
The gold medallist stands on the centre platform.
Therefore the height of the gold medallists’ platform is 1.2 m + 0.2 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’ platform.
Therefore the bottom of the structure supporting and forming the gold medalists’ platform consists of
two cuboids placed together with their square faces horizontal.
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The gold medallist stands on the centre
platform.
Seven cuboids are part of, or support, the
centre platform.
Therefore five cuboids are fitted together on top of these two base cuboids to form the centre platform.
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’ platform. (by visualising, sketching or making models of all possibilities)
The height of each cuboid is 0.4 m.
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.
The height of each cuboid is 0.4 m.
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 + 0.6 m = 1.4 m
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