A and B are playing the game of bulls and cows. The rules of the game are as follows.

A selects a word with distinct letters and B has to find it. B is allowed a maximum of ten guesses. After every guess A tells B the number of Bulls and Cows in the word.

Bulls are letters which appear in the correct location, and cows are letters which appear, but not in their correct location

For example, if A selects a word BOLD while B guesses LOUD. Then A's reply would be 2 Bulls and 1 Cow.(O and D are in the correct place while L is not). Note that A only tells the number of Bulls and Cows. He does not tells which ones are cows and bulls.

A selected a six letter word. The game proceeded as follows:

1. FRIEND - 0 Bulls, 2 Cows
2. SQUARE - 0 Bulls, 2 Cows
3. GROUND - 0 Bulls, 3 Cows
4. FORMAT - 1 Bull, 0 Cows
5. TRAVEL - 1 Bull, 1 Cow
6. FORMAL - 0 Bulls, 0 Cows
7. FLIGHT - 2 Bulls, 0 Cows
8. SYMBOL - 0 Bulls, 1 Cow
9. FIGURE - 0 Bulls, 3 Cows


What is A's word?

(Assume that A's word is a valid English word and has 6 distinct letters.)

From 4 and 6, the last letter is T, and there is none of the letters in FORMAL. The cow in 5 must be T, and the bull cannot be R, A, or L, so it is either V or E. One of the bulls in 7 is T. The other bull cannot be F or L. If the other bull in 7 is I, then I is the third letter, but that contradicts 1. If the other bull is H, then I and G are not in the word. Then, the only letters in FIGURE which can be in the word are U and E, which contradicts 9. Therefore, the bulls in 7 are G and T. Then, the bull in 5 cannot be V, so it is E. Therefore, the word ends in GET. By 9, the other cow is U. By 1, there is either an N or a D. By 2, there is no S or Q. By 8, there is either a Y or a B. The only word that works is BUDGET.

Posted by Math Man on 2013-06-04 21:45:42