A – Anne did it

B – Bill did it

C – Caroline did it

At least one of them did it:

A or B or C = true

Solving equations (AND):

| not A xor (not A and B and not C) = true

| not B xor C = true

| not C xor B = true

| A or B or C = true

Then

| not A xor (not A and B and not C) = true

| B = C

| A or B or C = true

Simplifying xor:

(A and (not A and B and not C)) or not A and (A or not B or C) = true

false or not A and (A or not B or C) = true

not A and (A or not B or C) = true

A = false

Remembering that

| B = C

| A or B or C = true

we get

false or B or B = true

B = true

C = true

Answer: Bill and Caroline

The argument for Anne’s innocence is valid. But we can’t deduce only from Anne’s statements that Bill did it. It does follow from Anne’s statements that *if* he did it, he didn’t do it alone. Caroline is still the only other person who could have done it. So Caroline definitely did it, either alone or with Bill.

Now can prove Bill’s involvement in either of the following two ways:

1) Bill’s statement that Caroline did it is true. Therefore his statement that he didn’t do it is false.

2) Caroline’s statement that she didn’t do it is false. Therefore her statement that Bill did it is true.

If Bill’s and Caroline’s statements had been thrown out, Caroline would have been implicated in the robbery, but not Bill.

]]>Thanks for a fun puzzle! I enjoy reading your blog.

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