55 lines
1.9 KiB
Markdown
55 lines
1.9 KiB
Markdown
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---
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id: 5900f46c1000cf542c50ff7e
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title: 'Problem 256: Tatami-Free Rooms'
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challengeType: 5
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forumTopicId: 301904
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dashedName: problem-256-tatami-free-rooms
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---
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# --description--
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Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
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Assuming that the only type of available tatami has dimensions 1×2, there are obviously some limitations for the shape and size of the rooms that can be covered.
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For this problem, we consider only rectangular rooms with integer dimensions a, b and even size s = a·b. We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition a ≤ b.
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There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet. For example, consider the two arrangements below for a 4×4 room:
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The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.
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Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms. Further, we define T(s) as the number of tatami-free rooms of size s.
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The smallest tatami-free room has size s = 70 and dimensions 7×10. All the other rooms of size s = 70 can be covered with tatami; they are: 1×70, 2×35 and 5×14. Hence, T(70) = 1.
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Similarly, we can verify that T(1320) = 5 because there are exactly 5 tatami-free rooms of size s = 1320: 20×66, 22×60, 24×55, 30×44 and 33×40. In fact, s = 1320 is the smallest room-size s for which T(s) = 5.
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Find the smallest room-size s for which T(s) = 200.
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# --hints--
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`euler256()` should return 85765680.
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```js
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assert.strictEqual(euler256(), 85765680);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler256() {
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return true;
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}
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euler256();
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```
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# --solutions--
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```js
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// solution required
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```
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