98 lines
7.5 KiB
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98 lines
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<h2>Tatami-Free Rooms</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 256</h3><span style="width:300px;color:#666;">Published on Saturday, 19th September 2009, 01:00 am; Solved by 466</span></div>
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<div class="problem_content" role="problem">
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<span style="font-size:10pt;">
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<p>Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.</p>
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<p>Assuming that the only type of available tatami has dimensions 1<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />2, there are obviously some limitations for the shape and size of the rooms that can be covered.</p>
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<p>For this problem, we consider only rectangular rooms with integer dimensions <var>a</var>, <var>b</var> and even size <var>s</var> = <var>a</var>·<var>b</var>.<br />
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We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition <var>a</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>b</var>.</p>
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<p>There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.<br />
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For example, consider the two arrangements below for a 4<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />4 room:</p>
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<img src="project/images/p_256_tatami3.gif" /><br />
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</div>
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<p>The arrangement on the left is acceptable, whereas the one on the right is not: a red "<span style="color:red"><b>X</b></span>" in the middle, marks the point where four tatami meet.</p>
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<p>Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms.<br />
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Further, we define <var>T</var>(<var>s</var>) as the number of tatami-free rooms of size <var>s</var>.</p>
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<p>The smallest tatami-free room has size <var>s</var> = 70 and dimensions 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10.<br />
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All the other rooms of size <var>s</var> = 70 can be covered with tatami; they are: 1<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />70, 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />35 and 5<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />14.<br />
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Hence, <var>T</var>(70) = 1.</p>
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<p>Similarly, we can verify that <var>T</var>(1320) = 5 because there are exactly 5 tatami-free rooms of size <var>s</var> = 1320:<br />
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20<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />66, 22<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />60, 24<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />55, 30<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />44 and 33<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />40.<br />
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In fact, <var>s</var> = 1320 is the smallest room-size <var>s</var> for which <var>T</var>(<var>s</var>) = 5.</p>
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<p>Find the smallest room-size <var>s</var> for which <var>T</var>(<var>s</var>) = 200.</p>
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