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Boxes of Matches (How Many Boxes? in the UK version) is a puzzle in Professor Layton and the Unwound Future.

## Puzzle

US Version

The two people shown below are talking about how many boxes of matches there are. The person on top (A) is looking down from above, and the person below (B) is looking at them from one side.

At most, how many boxes are there?

UK Version

Two people, A and B, are discussing the number of matchboxes on a table. A is describing the matchboxes as seen from above. B is describing them as seen from the side, but we don't know which side (front, back, left or right).

What is the maximum possible number of matchboxes on the table?

## Hints

Click a Tab to reveal the Hint.

US Version

Suppose B is looking at A's top-down image from the right. From B's side view, the most boxes would be, from left to right:

2 high x 2 stacks = 4 boxes
3 high x 3 stacks = 9 boxes
1 high x 3 stacks = 3 boxes
2 high x 2 stacks = 4 boxes

That's 20 boxes total. Is that the most?

UK Version

Take the layout that A described from above and imagine that B is looking at it from the right-hand side.
Working from left to right in B's description, each row of matchboxes would contain a maximum of:
2 piles x 2 layers
3 piles x 3 layers
3 piles x 1 layer
2 piles x 2 layers
So the maximum possible number of matchboxes in this case would be 20.

US Version

Suppose B is looking at A's top-down image from the bottom. From B's side view, the most boxes would be, from left to right:

2 high x 3 stacks = 6 boxes
3 high x 1 stacks = 3 boxes
1 high x 2 stacks = 2 boxes
2 high x 4 stacks = 8 boxes

That's 19 boxes total. Is that the most?

UK Version

Now take A's layout and imagine that B is looking at it from the bottom of A's top-down description.
Working from left to right in B's description, each row of matchboxes would contain a maximum of:
3 piles x 2 layers
1 pile x 3 layers
2 piles x 1 layer
4 piles x 2 layers
So the maximum possible number of matchboxes in this case would be 19.

US Version

Suppose B is looking at A's top-down image from the left. From B's side view, the most boxes would be, from left to right:

2 high x 2 stacks = 4 boxes
3 high x 3 stacks = 9 boxes
1 high x 3 stacks = 3 boxes
2 high x 2 stacks = 4 boxes

That's 20 boxes total. Is that the most?

UK Version

Thirdly, take A's layout and imagine that B is looking at it from the left.
Working from left to right in B's description, each row of matchboxes would contain a maximum of:
2 piles x 2 layers
3 piles x 1 layer
2 piles x 1 layer
2 piles x 2 layers
So the maximum possible number of matchboxes in this case would be 20.

US Version

Suppose B is looking at A's top-down image from the top. If you count them up as in the previous hints and there are more than 20, that's your answer.

Now all you need to do is count!

UK Version

Finally, take the layout that A described from above and imagine that B is looking at it from the top of A's top-down description.
Add up the number of matchboxes as shown in Hints 1-3. If this number is higher than 20, you know it's your answer.

Now all you have to do is count the matchboxes.

## Solution

### Incorrect

US Version

Imagine the arrangement of boxes from each angle: front, back, left, and right.

UK Version

Try to imagine looking at the matchboxes from every direction: front, back, left and right.

### Correct

US Version

Correct!

There are 21 boxes at most.

UK Version

Boxing clever!

21 matchboxes.
The above layout allows you to have the maximum possible number of matchboxes.

A big thanks to http://professorlayton3walkthrough.blogspot.com

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