### Puzzlepedia^{™} Math Puzzles

# Survo

Fill the grid with numbers to make the row and column sums add up.

In a Survo puzzle, the task is to fill a table with integers so that each of the integers appears only once and their row and column sums are equal to integers given on the bottom and the right side of the table. The smallest integer that can be used is 1 and the largest integer is equal to the number of cells in the table (which can vary). These are similar to Kakuro puzzles. In both types of puzzles you are fitting numbers into rows and columns to make the sums accurate. The differences are that in a Kakuro puzzle the geometry is more like a crossword puzzle and numbers can repeat within the grid.

**For Example:**

In the following sample puzzle there are three rows (labeled 1-3) and four columns (labeled A-D). This gives us 12 cells that we must fill with each of the numbers 1 through 12. Three of the cells have been given to get us started.

A good place to start solving these types of puzzles is to look for rows or columns that are partially solved, especially those where the sum is high or low, because this will give us the fewest possible combinations of numbers that could fit. Take a look at column B, which we know must add up to 16. We are given a 6 already, which means that the remaining two cells must add to 10. The only way we can do this, without repeating a number already placed into the grid is with 1+9. The 9 cannot be placed into row 2 because that would make it impossible to get its sum to equal 18, so the only solution to column B is as follows.

Now take a look at column A which sums to 27. The only possible missing numbers that work (without repeating any numbers in the grid) are 7+12. If the 12 is in row 3, then the last missing number in that row would have to be 6 to make the sum equal to 30, but we already have a 6 placed in the grid, so the 12 must go in row 1 and the 7 in row 3. This means that the last missing number in row 3 must be 11.

There are four empty cells left, and we know that they must contain the missing numbers 2, 4, 5 and 10. Column C requires a sum of 10, with a 3 already giving in this column. Therefore, the numbers 2+5 must be in column C and 4+10 must be in column D. By looking at the sums for rows 1 and 2 we can quickly arrive at the final solution.

If this type of deductive reasoning is not good enough to completely solve a Survo puzzle, the "swapping method" can be used. To use this method, you first randomly fill the empty cells with the remaining numbers. Now, compute the summation error for each of the rows and columns. For example, column B may be 2 over the intended sum and column C may be 3 under the intended sum. Now, find a row and column that are both under their intended sum and locate the cell that they have in common. Also find a row and column that are both over their intended sum and locate their intersecting cell. Swap these two cells and recompute the summation errors. This should get your errors closer to zero. By repeating this process several times, you can arrive at the solution.

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