Logical Larry was upset to learn that his recent submission to a logic problem magazine had multiple solutions. As it turns out, the magazine editor accidentally omitted one of his clues.

Here is the clue as it was printed:

Four children (Betty, Bobby, Billy, and Brian) enter a store selling bikes, balls, and balloons, each of which come in blue, brown, or beige.

From the clues below, determine what each child purchased:

1. Betty did not purchase a balloon.

2. Bobby and Billy purchased the same color item, which was not brown. Betty and Bobby purchased the same item.

3. Betty thought it best to purchase a brown item.

4. Brian is the only one who purchased both his item and his color.

5. A brown ball was not purchased. Neither was a blue bike.

The missing clue should have read, "A balloon was not purchased by _________" However, Larry cannot remember what should be in the blank to make the problem solvable.

What should be in the blank?

Although the pairings of colors, items, and people are not exclusive, as with most "clue-logic" puzzles, it may still be useful to organize the matchups in the grid. It simply means that all significant information may not be represented in the grid.

Since everything starts with B, we'll go by bEtty, bObby, bIlly, bRian, bIke, bAll, baLloon, bLue, bRown, and bEige to represent them.

  I A L  L R E
E _ _ _  _ _ _
O _ _ _  _ _ _
I _ _ _  _ _ _
R _ _ _  _ _ _

L _ _ _
R _ _ _
E _ _ _

Let's use X to represent things we know aren't true, and · to represent things we know are.

Now, going through the clues, we know that Betty did not buy a balloon (1).

  I A L  L R E
E _ _ X  _ _ _
O _ _ _  _ _ _
I _ _ _  _ _ _
R _ _ _  _ _ _

L _ _ _
R _ _ _
E _ _ _

That two people purchased items of the same type or color cannot be represented in the grid until we know what that item is (or is not). However, we do know from clue 2 that Bobby and Billy did not buy something brown:

  I A L  L R E
E _ _ X  _ _ _
O _ _ _  _ X _
I _ _ _  _ X _
R _ _ _  _ _ _

L _ _ _
R _ _ _
E _ _ _

Betty did purchase a brown item:

  I A L  L R E
E _ _ X  X · X
O _ _ _  _ X _
I _ _ _  _ X _
R _ _ _  _ _ _

L _ _ _
R _ _ _
E _ _ _

We can do nothing with clue 4 until we know what someone else has purchased. At this point, he did not buy something brown (Betty did).

  I A L  L R E
E _ _ X  X · X
O _ _ _  _ X _
I _ _ _  _ X _
R _ _ _  _ X _

L _ _ _
R _ _ _
E _ _ _

Now, we know that only Betty bought a brown item, and it was not a balloon (hence, no brown balloons were purchased).

  I A L  L R E
E _ _ X  X · X
O _ _ _  _ X _
I _ _ _  _ X _
R _ _ _  _ X _

L _ _ _
R _ _ X
E _ _ _

A brown ball was not purchased (5), so the brown item that Betty bought must have been a bike. Also, since Betty bought a bike, so did Bobby (2), but not Brian (4):

  I A L  L R E
E · X X  X · X
O · X X  _ X _
I _ _ _  _ X _
R X _ _  _ X _

L _ _ _
R · X X
E _ _ _

A blue bike was not purchased (5), so Bobby's bike must be beige. Thus, Billy bought something beige as well (2). Also, Brian didn't buy the same color item as they did (4), so his was blue:

  I A L  L R E
E · X X  X · X
O · X X  X X ·
I _ _ _  X X ·
R X _ _  · X X

L X _ _
R · X X
E · _ _

Now, here's what we know:

Betty bought a brown bike.
Bobby bought a beige bike.
Billy bought a beige something.
Brian bought a blue something, but not a bike, and not the same thing as Billy did.

So, the last clue must determine something about Billy and Brian.

If it reads, "A balloon was not purchased by Billy," then Billy could have bought a bike or a ball. In either case, Brian could then have bought the balloon, or a ball if Billy got a bike.
In other words, that wouldn't help us.

If, however, the clue read, "A balloon was not purchased by Brian," then of course Brian is left buying a blue ball. Yeah. But Billy could still be buying a beige bike, just like his big buddy Bobby, or a beige balloon (unappealing as that seems, it's still possible).
If the clue had said that no two people bought exactly the same item and color, then filling in the blank with "Brian" would indeed lead to a unique solution.

So, only one option remains. The clue was supposed to read: "A balloon was not purchased by anyone" (it never said the blank was filled with a name). Thus, Brian is left, again, buying a blue ball, and Billy has to be behind Bobby and buy a brand new beige bicycle.

So, if nobody bought a balloon, here's what we have:

  I A L  L R E
E · X X  X · X
O · X X  X X ·
I _ _ X  X X ·
R X _ X  · X X

L X _ X
R · X X
E · _ X

As stated above, Brian bought a ball, so Billy didn't; he bought a bike.

  I A L  L R E
E · X X  X · X
O · X X  X X ·
I · X X  X X ·
R X · X  · X X

L X · X
R · X X
E · X X

Boy, oh boy ..

Posted by DJ on 2004-04-27 15:10:14