jmc

Question

In a rectangular array of points, with 5 rows and N columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through N, the second row is numbered N + 1 through 2N, and so forth. Five points, P_1, P_2, P_3, P_4, and P_5, are selected so that each P_i is in row i. Let x_i be the number associated with P_i. Now renumber the array consecutively from top to bottom, beginning with the first column. Let y_i be the number associated with P_i after the renumbering. It is found that x_1 = y_2, x_2 = y_1, x_3 = y_4, x_4 = y_5, and x_5 = y_3. Find the smallest possible value of N.

Accepted Answer

Let each point P_i be in column c_i. The numberings for P_i can now be defined as follows.align*x_i &= (i - 1)N + c_i y_i &= (c_i - 1)5 + i align* We can now convert the five given equalities.alignx_1&=y_2 & & & c_1 &= 5 c_2-3 x_2&=y_1 & & & N+c_2 &= 5 c_1-4 x_3&=y_4 & & & 2 N+c_3 &= 5 c_4-1 x_4&=y_5 & & & 3 N+c_4 &= 5 c_5 x_5&=y_3 & & & 4 N+c_5 &= 5 c_3-2 alignEquations (1) and (2) combine to formN = 24c_2 - 19Similarly equations (3), (4), and (5) combine to form117N +51 = 124c_3Take this equation modulo 3124N+20 0 31And substitute for N24 24 c_2 - 24 19 +20 0 3118 c_2 2 31 Thus the smallest c_2 might be is 7 and by substitution N = 24 7 - 19 = 149 The column values can also easily be found by substitutionalign*c_1&=32 c_2&=7 c_3&=141 c_4&=88 c_5&=107 align*As these are all positive and…