For a fistful of dollars
Aditya Rishi

THE Magnificent Seven pull the triggers together, but can’t quite bring themselves to kill, which can be dangerous because their opponent is armed and he’ll, now, shoot back.

Two weeks ago, the challenge was to find out how the good, the bad and the ugly solved their dispute in the Wild West. The good possessed seven English horses, the bad had nine Arabian horses and the ugly had 10 camels. Each gave two animals, one to each of the others. They were, then, equally well off. The Sheriff of El Paso was in search of cowboys who could find the price of each animal and the total value of the animals possessed by each person. He found seven, but not all were good.

The solution, translated into mathematical notation, proceeds as follows: the cowboys seek integer solutions x1, x2, x3 and k (where x1 is the price of an English horse, x2 is the price of an Arabian horse and x3 is the price of a camel) satisfying

5 x1 + x2 + x3 = x1 + 7 x2+ x3 = x1 + x2 + 8 x3 = k.

Then, 4 x1 = 6 x2 = 7 x3 = k - (x1 + x2 + x3).

For integer solutions, k - (x1 + x2 + x3) must be a multiple of the LCM of 4, 6 and 7. Taking different multiples of the LCM will lead to different solutions. The good, perhaps, solved the system of three equations for x1, x2, x3 in terms of k to obtain x1 = 21k/131, x2 = 14k/131, x3 = 12k/131.

He obtained integer solutions by taking k = 131, which is the smallest solution, achieved also by taking k - (x1 + x2 + x3) = LCM (4, 6, 7) = 84.

Here’s what Gringo Dr Tarsem Lal has to say about this: "At the end of good’s formula, good is left with 5 good English horses, one Arabian horse and one camel, bad’s property is 7 bad Arabian horses, one English horse and one camel, and ugly has 8 ugly camels, one English horse and one Arabian horse. If, for the sake of brevity, we denote an English horse with ‘E’, an Arabian horse with ‘A’ and a camel with ‘C’, then, we have:

5E+1A+1C=7A+1E+1C=8A+1E+1A

A little shifting of elements in the above equations will show that 4E=6A and 6A=7C and LCM of 4,6 and 7 is 84, so, E=21 units of currency, A = 14 units of currency and C= 12 units of currency; so, each one has 131 units of currency."

Gringo Tarsem Lal gets the Sheriff’s badge of honour. The other good shooters in the squad are Neha Goyal, Priyanka and Kamal Sethi. The shot fired by Mansha Singh kissed the Sheriff’s hat. That was close. Ajit Partap Singh can tell us about McKenna’s Gold, but cannot lead us to it. Divya got excited and shot wide, but her way of shooting is original. She’s the kid of the squad, but, perhaps, we underestimate her aim. (Write at The Tribune or adityarishi99@yahoo.co.in)