The Contest Center - SumSum Puzzle

SUMSUM^TM

       SumSum is a new kind of puzzle that uses numbers and logic. SumSum is like a crossword puzzle that uses numbers instead of letters. There is a grid where you can write the numbers. Your job is to figure out which numbers to write in which boxes. You do that by using the clues.

       Each puzzle will have some rules telling you what numbers to use. For instance, the rules might say "Write the numbers from 1 to 5." You would write the numbers from 1 to 5 into 5 of the boxes in each line. A line can be a row or a column. Rows read across and columns read down.

       That will leave 2 boxes over. You can fill in the other 2 boxes solid black, like a crossword puzzle. The black boxes split each line into several sections called runs. There can be 1, 2 or 3 runs in a line. The two black boxes in a line never touch each other, but they may touch the black boxes from other lines.

CLUES

       To the left of each row is a wide box with the clues for that row. The clues tell you the sum of the numbers in each run. The clues are given in order, from left to right. In this example the clues are 5, 1 and 9.

       The first run is 2 and 3. The first clue is 5, which is 2+3. The second run has the single number 1, so the second clue is 1. The third run is 5 and 4. The third clue is 9, which is 5+4.

5,1,9

It works the same way in the columns, going from top to bottom.

       Now that you know all about the grid and the clues, you are ready for

SOME SUPER SUMSUM SOLVING SECRETS

       You can be a SumSum genius if you know a few simple tricks. The clues to the left of each row and at the top of each column tell you what numbers to use. For example, if a clue is a 1 or a 2, then you will use a single 1 or a 2 because there is no way for two different numbers to add up to 1 or 2.

       The number of clues tells you which boxes must be black and which boxes must be white. Some black boxes can be placed immediately. If a line has only one clue, then the two end boxes must be black. All the rest of the boxes must be white.

       This line has only one clue, which is 15. So you know that the two black boxes must go at the ends of the line. You can mark them black by filling them in. You can mark boxes that must be white with a dash, like this

If a line has three clues, then both end boxes must be white. This example has three clues, 5, 6 and 4. So you can mark the two end boxes white, like this

5,6,4

If a line has two clues, then one end box must be white and the other end black. If you know that the box at one end is white, then you can mark the other end black. If you know that the box at one end is black, then you can mark the other end white.

The two black boxes in any line can't touch. So, every time you mark a box black, you can mark the surrounding boxes white, like this

		-			-
1,12,2	1		-	-		2
		-			-

When there are only two boxes left in a line that don't have to be white, those last two boxes have to be black.

Once you have found where the two black boxes go, you can mark all the other boxes in that line white, like this

5,8,2

After you have found some of the black boxes, start looking for places where there is just a single white box. The clue will tell you what number to put there. In both of these examples, the number in the yellow box has to be 4.

6,4,5

5,5,5

LET'S TRY AN EXAMPLE

Take a look at this line from a SumSum puzzle. This puzzle uses the numbers from 1 to 5.

1,8,6

Since there are 3 clues on this line, the first box must be white. The first clue is 1. The only way that you can make 1 using the numbers 1 to 5 is with the single number 1. This tells you that the first box must contain the number 1, and the second box must be black.

1,8,6

This leaves 5 more spaces. The second clue is 8. It will take at least 2 numbers to add up to 8. You can mark the next two boxes white, like this

1,8,6

The third clue is 6. It will also take at least 2 numbers to add up to 6. You can mark the last two boxes white, like this

1,8,6

That leaves only one more box. The last box has to be black, like this

1,8,6

You can finish the puzzle by doing the same thing in the other lines.

THE WHOLE THING

Now that you know how to tackle each line, you are ready to try a complete puzzle. Here is a puzzle from the SumSum puzzle website. This puzzle uses the numbers from 1 to 5.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6
3,6,6
8,5,2
2,13
9,6
8,4,3
1,14

A good place to start is to mark the boxes that have to be white. Remember, if there are 3 clues for a row, then the first and last boxes have to be white. If the clue is more than 5, then there have to be at least 2 white boxes.
Let's first use the row clues to find some boxes at the ends of the rows that have to be white. Each time a box changes that will shown in yellow.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6
3,6,6	-					-	-
8,5,2	-	-					-
2,13
9,6
8,4,3	-	-					-
1,14

Whenever a clue is a 1 or a 2 that run can have only one number. You can write that number right away, and you can mark the next box black.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6
3,6,6	-					-	-
8,5,2	-	-					2
2,13
9,6
8,4,3	-	-					-
1,14

All of the boxes next to a black box must be white.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6
3,6,6	-					-	-
8,5,2	-	-			-		2
2,13						-
9,6
8,4,3	-	-					-
1,14

Now let's do the same thing for the columns.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6		-		-	2	-
3,6,6	-					-	-
8,5,2	-	-			-		2
2,13						-
9,6
8,4,3	-	-					-
1,14		-		-	-	2

When a line has only 1 clue, both end boxes have to be black, and the 5 middle boxes must be white. You can mark all of the boxes in column 3.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6		-		-	2	-
3,6,6	-		-			-	-
8,5,2	-	-	-		-		2
2,13			-			-
9,6			-
8,4,3	-	-	-				-
1,14		-		-	-	2

Look at the top row. The first run totals 9. You need at least 2 numbers to make 9, so the upper left corner must be white. There are 2 clues for the top row, so if the first box is white, the last box must be black. The same for the first column. So the upper right corner and the lower left corner must be black. That means the lower right corner is white. That finishes all 4 corners.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	-		-	2	-
3,6,6	-		-			-	-
8,5,2	-	-	-		-		2
2,13			-			-
9,6			-
8,4,3	-	-	-				-
1,14		-		-	-	2	-

Now you can fill in a few of the numbers by sight.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	-		-	2	-
3,6,6	-		-			-	1
8,5,2	-	-	-		-		2
2,13			-			-
9,6			-
8,4,3	-	-	-				3
1,14		1		-	-	2	-

That makes it clear where some more of the black and white boxes have to be.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	-		-	2	-
3,6,6	-		-	-		-	1
8,5,2	-	-	-		-		2
2,13			-			-
9,6			-				-
8,4,3	-	-	-		-		3
1,14		1		-	-	2	-

Now there are a few more single numbers that can be filled in by sight.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	5		-	2	-
3,6,6	3		-	-		5	1
8,5,2	-	-	-		5		2
2,13			-			-
9,6			-				-
8,4,3	-	-	-		4		3
1,14		1		5	-	2	-

Row 4 has just 2 clues. Since the last box is black the first box must be white. Row 5 is exactly the opposite.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	5		-	2	-
3,6,6	3		-	-		5	1
8,5,2	-	-	-		5		2
2,13	-		-			-
9,6			-				-
8,4,3	-	-	-		4		3
1,14		1		5	-	2	-

In column 4 and column 6 both of the black boxes have been marked. All the other boxes in those columns must be white.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	-	5		-	2	-
3,6,6	3		-	-		5	1
8,5,2	-	-	-		5		2
2,13	-		-	-		-
9,6			-	-		-	-
8,4,3	-	-	-		4		3
1,14		1		5	-	2	-

Now more of the missing numbers can be filled in.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	-	-	-		5		2
2,13	2		-	-		-
9,6			-	-		-	-
8,4,3	5	-	-		4		3
1,14		1		5	-	2	-

On row 6 the 2 missing numbers must be either 1,2 or 2,1. But there is already a 1 in column 2. That means the 2 missing numbers must be 2,1 in that order.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	-	-	-		5		2
2,13	2		-	-		-
9,6			-	-		-	-
8,4,3	5	2	1		4		3
1,14		1		5	-	2	-

That lets you fill in the rest of column 1 and column 2. There is only one place left for the last black box, so you can mark that, too.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	1	4	-		5		2
2,13	2		-	-		-
9,6		3	-	-		-	-
8,4,3	5	2	1		4		3
1,14		1		5	-	2	-

You can now finish row 3 and column 5.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	1	4	3		5		2
2,13	2		-	-	1	-
9,6		3	-	-		-	-
8,4,3	5	2	1		4		3
1,14		1		5	3	2	-

You can now finish row 7. That will let you fill in some more numbers.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	1	4	3		5		2
2,13	2		-	-	1	4
9,6		3	-	-		1	5
8,4,3	5	2	1		4		3
1,14		1		5	3	2	4

The last 2 numbers on row 4 can only be 3,5 or 5,3. Since there is already a 3 in column 3, the only choice is 5,3. That gives you the last 2 numbers, and the puzzle is finished.

SumSum	10 5	5 4 6	15	3 7 5	2 6 7	8 5 2	3 12
9,6	4	5		1	2	3
3,6,6	3		4	2		5	1
8,5,2	1	4	3		5		2
2,13	2		5	3	1	4
9,6		3	2	4		1	5
8,4,3	5	2	1		4		3
1,14		1		5	3	2	4