Solving Jasper Tsang's Math Riddle

Legco Chairman Jasper Tsang set a math riddle that he claimed would encode the identity of the next Chief Executive of Hong Kong1.

A forest was inhabited by three kinds of creatures: pheasants, wolves and nine-headed birds. The arithmetic sum of their heads is 100, and so is the sum of legs. After some time, their populations shrank a bit, but if each creature had one more head and one more leg, the arithmetic sum of heads and that of legs were still 100 each. What was the minimum total number of creatures still present in the forest? The answer, according to Tsang, tells who the next Chief Executive will be.

Various solutions have been posted since Tsang published his riddle. Some are incredibly complex and some rather heuristic. Here I am offering a quick but rigorous solution. Let x, y, and z be the number of pheasants, wolves and nine-headed birds, respectively. Also denote the total number by N. We know that a pheasant has 1 head and 2 legs, a wolf has 1 head and 4 legs, and a nine-headed bird presumably has 9 heads and 2 legs. The situation where each creature has one more head and one more leg can be represented by the following set of equations:

2x + 2y + 10z + 0 = 100 (total number of heads is 100)

3x + 5y + 3z + 0 = 100 (total number of legs is 100)

x + y + z – N = 0 (total population)

This is a set of 3 equations with 4 unknowns. In theory, there could be infinite possible solutions. High-school students may try to solve it by elimination, but a handy process (technically called Gauss-Jordan reduction process) would lead conveniently to the following solutions:

x = (11N – 250) ÷ 4

y = (100 – 3N) ÷ 2

z = (50 – N) ÷ 4

Clearly, I can choose any N, and generate as many solutions as I wish. But since only non-negative whole numbers (integers) are allowed, the above equations already tell us that N must lie between 23 and 33. Why? The first equation says 11N > 250, and hence N ≥ 23; and the second equation says 100 > 3N which means N ≤ 33. Let's try each possible whole number for N one by one, and it takes just a few minutes to figure out N = 30, x = 20, y = 5, and z = 5.

So, who is the next Chief Executive? The cipher is 30. I suppose it's up to anyone to decode it with any conjectural connection one thinks plausible2. I am sure Tsang does not know the answer but set the riddle to probe informally into the minds of the public; otherwise it is nothing but a typical fortune teller's trick of the trade. My own focus of Tsang's indefinite cipher is on the population of individual creatures: 20 pheasants, 5 wolves and 5 ten-headed birds! The pheasant would lead the wolf over a wide margin! That'll be the most wishful for Hong Kong, regardless of who the pheasant represents.

July 2016

Standard shortcut developed by Gauss and modified by Jordan back in 1888

The three equations can be transformed to a set of 15 numbers put in the following format (called matrix in mathematics).

The sequence of matrices is obtained by repeatedly adding/subtracting one row to/from another row, generating a new row, and you may scale up the whole row by any fixed ratio before adding and subtracting. In the process, you need to add and subtract cautiously aiming to produce rows that comply with a specific format (technically called reduced echelon form) containing mostly zeros similar to the very bottom matrix shown above, from which you can read off the answers directly. For example, the first row says x – 11N/4 = –125/2. Likewise, y and z can be found in terms of N.