2013/round3/digits_war.html

<p>There has been a war between the digits in the kingdom of numbers and it is King Infinity's job to restore balance. In search of peace he came up with a new number system which only allows those numbers to exist in which:
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<p>
1. None of the consecutive digits are at war against each other. 
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<p>
2. No two digits that have only one digit in between them are at war.
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<p>For example, if 4 is at war with 5, then 45, 405, and 574 are all forbidden.</p>
<p>
A digit can be at war with itself. You are given a 10 x 10 binary matrix <strong>M</strong> (0 index based), where <strong>M</strong>[i][j] denotes whether there is a war between digit i and digit j. If <strong>M</strong>[i][j] = 1 then they are at war and <strong>M</strong>[i][j] = 0 means they are not. <strong>M</strong>[i][j] will always be equal to <strong>M</strong>[j][i].
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<p>
Your task is to find the count of positive numbers that can exist in this number system with number of digits &le; <strong>K</strong>. No number in the number system can have leading zeroes.
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<p>
<h2 id="limits">Limits</h2>
1 &le; <strong>K</strong> &le; 10<sup>18</sup></h2>
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<p>
<h2 id="input">Input</h2>
Input consists of <strong>T</strong> test cases, with <strong>T</strong> &le; 25. Each test case begins with the value of <strong>K</strong> followed by a 10x10 binary matrix.
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<p>
<h2 id="output">Output</h2>
For every test case output the result modulo 10<sup>9</sup> +7 
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