Question 1.4 - Number the 12 edges of a cube with numbers 1 through 12 in such a

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Question 1.4 - Number the 12 edges of a cube with numbers 1 through 12 in such a way that the sum of the three edges meeting at each vertex is the same for each vertex.

1.2 Suppose we are given n disjoint sets Si. Let the first have ai elemen ond a2, and so on. Show that the number of sets that contain at most one from each S, is equal to (a1 + 1) a). ts,t sec +1). Apply this result to the following number theoretic problem: Let n = r" p be the prime decomposition of n. Then n has exactly t(n) II(ai 1) divisors. Conclude that n is a square if and only if t(n) is odd. Let N = { 1, 2, . . . , 100, and let A be a subset of N with IA-55. Show that A contains two numbers a and b such that a -b 9. Does this hold as well for |A] = 54? 1.4 Number the twelve edges of a cube with the numbers 1 through 12 in such a way that the sum of the three edges meeting at each vertex is the same for each D> 1.3 vertex. D 1.5 In the parliament of country X there are 151 seats and three political parties. How many ways (i, j, k) are there of dividing up the seats such that no party has an absolute majority? 1.6 How many different words can be made from permutations of the letters in ABRACADABRA? 1.7 Show that 1!+2!+ + n! for n > 3 is never a square. 1.8 Show that for the binomial coefficients (H) the following holds:

Explanation / Answer

Suppose we have done such a numbering. Let x be the sum of three of edges meeting at each vertex.

Notice that each edge is connected to two vertices, that is each edge contribute to the sum at two vertices.

Thus 8x = twice the sum over all the edges (every edge is counted twice).

That is 8x = (1+2+3+....+12)*2 = 156.
Thus x = 19.5
Now sum of positive integers is always a positive integer, but x is not. So we cannot arrange numbers 1 to 12 in a way such that the sume is 19.5 at each vertex.

Thus it follows that the problem has no solution.

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