Faulty Symmetry

The Math Horizons magazine (v 17, n 4, April 2010, pp. 31-33) published a solution to Problem 239 that was posed by former problem section editor Andy Liu, who found inspiration for the problem from the book Inversions by Scott Kim. Doubled Over asked readers to find an integer root of each of the following two equations:

(a)
9 (8 - x)

(9 - 8) x
+
8 - x

9 - 8
+
11 - x

x - 1
x
(b)
x  =
1 - x

x - 11
+
8 - 6

x - 8
+
x (8 - 6)

(x - 8) 6

Equation (a) is easily converted to a polynomial equation:

  x³ - 42x + 36 = 0.

According to the rational root theorem, all integer roots of the equation must divide the free term, 36 in this case. The latter has the following factors

  ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36.

It takes a little time to check that only x = 6 satisfies the equation.

Reaching this point the problem editor made a remark:

  The same type of analysis could be applied to equation (b). But James Hochschild and Zhiheng Li both noticed a very useful symmetry: equation (b) is equation (a) rotated by 180 degrees! Thus, we may simply rotate the solution to (a) by 180 degrees to arrive at 9 = x, the solution for (b).

This argument is a very hilarious mathematical joke. Given that it appeared in the April's issue of the magazine, I am inclined to think that this is what it meant to be. Lest it be perceived otherwise, I offer a couple of simplified equations that follow the ingenious Andy Liu's exercise:

(a')
x - 1

11 - x
+
8 - x

9 - 8
+6
x
(b')
x  =
9+
8 - 6

x - 8
+
x - 11

1 - x

As can be easily seen, x = 9 solves (a') while x = 6 does not fit into (b'). The wink at the symmetry for solving (b) had to be an April's Fool's joke, for, otherwise, it would serve an example of a faulty use of symmetry. Trying to come up with a pair of equations in the spirit devised by Andy Liu, one can't help but marvel at his ingenuity.

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