Why is it important that the digits be all different?

Question

In standard Bulls and Cows, the digits of the chosen number must be all different.

Is this rule crucial for the "playability" of the game? In other words: if I ignore this rule and allow choosing a code with some identical digits - does it make the game to hard or too easy for the guesser?

Answer 1

As per your linked wikipedia article, a later game based on the same concept is Mastermind. And from that game's Wikipedia article:

Duplicates and blanks are allowed depending on player choice, so the player could even choose four code pegs of the same color or four blanks.

So no, it's not crucial. It makes the game harder in that there are more possible solutions.

Answer 2

Allowing duplicates makes the game more difficult for the guesser in two ways:

1. As notes in other answers, there are more possible solutions (10,000 versus 5,040).

2. Each "hint" from the mastermind gives potentially less information.

There would be a few ways to deal with the rules, but for simplicity sake let's assume that the mastermind must answer "correct position" if the peg's colour is indeed in the same position, and each peg in the guess can only clue one peg in the solution. For example, suppose the solution is WWGB. Then:

1. A guess of YYBB would get one red peg (correct colour, correct position).

2. A guess of YYBY would get one white peg (correct colour, wrong position).

3. A guess of WYYY would get one red peg, as would YWYY.

4. A guess of YYWY would get one white peg.

5. A guess of YYWW would get two white pegs.

6. A guess of YWWY would get one red peg, one white peg.

In standard mastermind, if you know that the first peg is white then you don't need to guess white for any other position. But with duplicates allowed, you can't make that assumption - you potentially need to guess WWWW to see how many whites are in the solution, then GGGG to see how many greens, and so forth.

Answer 3

The main reason for that rule is it makes it about twice as difficult if you allow for duplicate numbers instead of not allowing the.

Duplicates Not Allowed

• 10 Choices for first pick
• 9 choices for second pick
• 8 choices for third pick
• 7 choices for fourth pick

Total possible choices: 5040

10 * 9 * 8 * 7

Duplicates Allowed

• 10 Choices for first pick
• 10 choices for second pick
• 10 choices for third pick
• 10 choices for fourth pick

Total possible choices: 10000

10 * 10 * 10 * 10

The other part where it gets tricky is how you properly annotate what numbers that they have correct and in the correct spot versus what numbers they have correct but in the wrong spot