As stated above, a unique feature of this game is that a player’s counters are placed on the board by his opponent. One player will be “fire” and the other player will be “ice”.
Players then choose which color of counters will be theirs. The game commences with the board empty of counters. The first player to control three islands connected by a line or circle wins the game. Note: The pattern of seven points, which are connected in triplets in seven ways, is called a Fano Plane, an example of the mathematical structure known as a "finite plane." Each point on the plane is mathematically identical to the others.įire & Ice was designed by Jens-Peter Schliemann and Published by Out of the Box Pin International. The board is made up of seven triangular “islands” and each island in turn has seven positions for counters. A unique feature of this game is that a player’s counters are placed on the board by his opponent. Alternatively, and even simpler, we can obtain the same identity from the coordinates of B which are $x,y = r\times 2,r \times \sqrt 2$.Fire & Ice commences with the board vacant of counters.Ī Fire & Ice board and twenty-five each of black and white counters (or red and aqua counters, holding more true to the original publication) are required for play. Applying Pythagoras and taking advantage of the mixed terms in the squares cancelling we get $(100 \mathrm m)^2 = r^2 \times 2 \times (2+1)$. Using some of the eight congruent right triangles indicated in orange we find: The coordinates of A relative to O are $x,y = r \times (\sqrt 2 + 1), r \times (\sqrt 2 - 1)$. It follows that the distance OX is $\sqrt 2 r$. Taking the mirror image wrt M therefore leaves O where it is and swaps X and Y. M is an axis of symmetry for both the adult's pool and the (full) circle. Let Y be the point where the two pools touch and r be the side length of the children's pool.Ĭonsider the perpendicular bisector M of A and B. Let O be the centre of the circle, A,B ccw the two points where the adults' pool touches the circle's perimeter and X the point where it touches the straight enclosure.
The puzzle boils down to: Two squares are inscribed in a unit quadrant as shown in this simplified diagram: what is the blue square’s area? While it is simple and straightforward, and has several possible methods of solution (any of which are welcome), and the answer is not unexpected, there is a simple elegant way to solve it, which is (in my opinion) nice enough to qualify it for this site, rather than being just a maths problem. The setup is flavourtext and non-essential. This is a purely geometrical puzzle of my own construction. How many hectares does the children’s pool occupy? The architect says their specifications left him no choice. "Exactly how many hectares is this pool?" they ask. The pools must meet at their corners so people can go between them easily.Īfter seeing the architect’s design (below), the grinchy councillors think the children’s pool is too big. Half the children's pool's perimeter must touch the edge, where a stand (L-shape) will be built for parents and lifeguards. They want to utilize all the suitable land they can, so the adults’ pool must reach the suitable land's edge at three of its corners, where the mandatory three lifeguards will be stationed (dots). They want a children's wade pool and a deeper adults' pool, both square. A one hectare ( $100\times100$m) block of land has ground suitable for pool-digging, but a local ring road runs through it, leaving only an exact quarter-circle available. The local council is building a new pool complex.
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