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Sweet Sums is a puzzle in Professor Layton and the Last Specter.

## Puzzle

US Version

"My children have been very kind to each other lately, so I decided to give them some candy as a reward.

I have four jars of candy, A, B, C, and D. The combined number of candies in jars A and B is equal to twice the number in jar C. The combined number of candies in jars B and D equals twice the number in jar A. If you take three candies from jar D and put them in jar A, jar A will contain twice the number in jar B.

Which jar contains six pieces of candy?"

UK Version

Below are four rather special sweetie jars, A to D.

The number of sweets in jars A and B combined is exactly twice the number of sweets in jar C. At the same time, the total number of sweets in jars B and D is twice the number of sweets in jar A, but if you move three sweets from jar D to jar A, jar A will have twice as many sweets as jar B.

One of the jars contains six sweets. Can you work out which?

## Hints

Click a Tab to reveal the Hint.

US Version

All you need to do is figure out which jar contains six pieces of candy.

Even if you can't work it out using complicated mathematical calculations, you should still be able to solve it by figuring out an alternate method for determining the number of candies in each jar.

UK Version

The only thing you need to work out in this puzzle is which of the jars contains six sweets.

If you aren't comfortable solving complicated equations to find the number of sweets in every jar, there's a simple way to differentiate between the contents of the jars and get your answer.

US Version

Here's the puzzle in equations:

A + B = 2C
B + D = 2A
A + 3 = 2B
D > 3

This might help you spot a detail that you may have missed.

UK Version

Try turning the problem into equations:

A + B = 2C
B + D = 2A
A + 3 = 2B
D > 3

US Version

A + 3 = 2B

2B must be an even number 2 is one of its factors. That means A + 3 is also an even number.

Subtracting 3 from an even number will always result in an odd number, which means A must be odd.

So if A (odd) + B = 2C (even), then what is B, an odd or even number?

UK Version

Take the equation A + 3 = 2B.
2B must be an even number because 2 is one of its factors. That means A + 3 is also even. Taking away 3 from an even number will give you an odd number, and therefore A must be odd.

So, if A (odd) + B = 2C (even), then what sort of number is B? Odd or even?

US Version

Odd + Odd = Even
Odd + Even = Odd

From this simple rule, you can figure out that number of candies in jar B must be odd that the number in jar D must also be odd.

This means that all of the jars except C contain an odd numbers of candies. Therefore, the number of candies in jar C must be...

UK Version

You know that the number of sweets in jar A is odd.

Odd + Odd = Even
Odd + Even = Odd

By applying this simple rule, you can deduce that jars B and D both contain an odd number of sweets.

In other words, every jar except jar C contains an odd number of sweets...

## Solution

### Incorrect

Think it through and try again.

### Correct

Correct!

You know that A + 3 = 2B. Now, as 2B must be even and 3 is odd, A must also be odd.

Similarly,
A (odd) + B = 2C (even), so B = odd
B (odd) + D = 2A (even), so D = odd

The jar you are looking for contains an even number of sweets. As C is the only remaining possibility, it must be the answer by default.

A big thanks to http://professorlayton4walkthrough.blogspot.com

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