## The Halfsies

Detective Radstein is investigating a robbery. He apprehends three suspects: Anne, Bill, and Caroline. The detective knows that no one else could have participated in the robbery. During the interrogation the suspects make the following statements:

- Anne: I didn’t do it. Bill did it alone.
- Bill: I didn’t do it. Caroline did it.
- Caroline: I didn’t do it. Bill did it.

Detective Radstein also discovered that all three suspects are members of
a club called *The Halfsies*. Every time they speak, they make two statements,
one of which is a lie and the other one is true. Who committed the
robbery?

## Marvin Littman:

Bill and Caroline did it together.

2 February 2019, 2:31 pm## Matthew Vandermast:

If Anne did it, then both of her statements are false instead of just one. So Bill did it, but he didn’t do it alone. Caroline is his only possible accomplice. So Bill and Caroline did it.

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

2 February 2019, 2:42 pm## Matthew Vandermast:

P.S. Bill’s and Caroline’s statements were thrown out because they hadn’t been Mirandized. But the beautiful thing is that only Anne’s statement was necessary.

2 February 2019, 2:49 pm## Matthew Vandermast:

P.P.S. I knew I was tired yesterday, but didn’t realize how much.

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.

3 February 2019, 6:57 am## Ivan Yugov:

In Russia, it’s school informatics, 10 form (“profile” level). Long but universal.

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):

12 February 2019, 10:36 pm| 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