## Introduction

sumpuzzle is a challenging test of logic and simple arithmetic in which the player positions numbers to satisfy six intersecting sums. These intersecting sums contain exactly 9 numbers: the digits 1-9. The starting puzzle presents a grid showing the intersecting sums and their totals, but not the numbers used. The player must work out where to place the numbers 1-9, each of which must appear exactly once.

## Sumpuzzle Example

Figure 1. is a screenshot from sumpuzzle showing the beginning of a game. Here only the operators +, -, x and / and the results are shown. The values that make up the sums are missing and each of their squares contains tiny "candidate" numbers 1-9. These are buttons which can be left clicked for removal or middle clicked for setting the number for the square. A right click will replace a deleted candidate. Also seen in this figure is the fact that the player has left clicked on the green icon at the end of the second horizontal sum. This has brought up a list of the 4 possible sets of numbers which satisfy that sum. The player hence knows that the first number in this sum is 5, 6, 8, or 9, that the second is 9, 8, 6 or 5, and the third is 8 or 5. Click to see an animation showing the use of the buttons to solve this complete puzzle. The delay between images is 2 seconds.

## Summary

The grid contains 6 simple sums: 3 across and 3 down. Initially only the operators (+ add, - subtract, x multiply, / divide) and the result of each sum are shown. Players fill in the numbers 1 to 9 which satisfy the intersecting sums. Each number only appears once in the grid. Arithmetic is performed left to right. The candidate numbers are shown in each of the 9 squares. Left-clicking on a candidate will erase it; middle-clicking will reveal it; right-click will set it as the solution for that square. The green wand icons at the end of each sum provide assistance: left-click will reveal a list of the sets of numbers which satisfy the sum (please see Figure 1.); right-click will remove any candidates that are not in the list; middle-click will restore all the candidates for the sum. At the top of the display is a Toolbar. The jigsaw piece button at the left is a menu which has options for entering puzzles, reading them from files or saving them to disk. The menu includes options for configuring the display and saving the current settings. It also has an option to set the number of sets of numbers that satisfy each sum: roughly speaking, the more sets the harder the puzzle. Next to the jigsaw icon is a calculator-like icon which is used for selecting a random puzzle; a thumb for checking and correcting errors, and a sad smiley for showing the solution. To the right is a clock which ticks every 5 seconds. Eack puzzle is generated when the player clicks on the calculator icon: there are an unlimted number of puzzles.

## Playing

Each of the squares which have yet to be given a value contain tiny "candidate" numbers 1-9. These are buttons which can be left clicked for removal or middle clicked for setting the number for the square. A right click will replace a deleted candidate. In this way they can be used as a simple notepad while working towards the solution for each square. If the player cannot make progress, hints are available.

## Using Hints

Hints are requested for individual sums using the wand icon at the sum's end. A left-click will bring up a list of all the possible sets of digits which satisfy the sum. A right-click will delete all candidates in the sum which do not appear in these sets. A middle-click will resore them.

Of course, use of the hints is not obligatory and purists can work out everything for themselves and simply use the candidate buttons as a notepad for each square.

## Loading Puzzles

The Puzzle menu contains options for reading puzzles from files, saving puzzles to files and for entering puzzles through an on-screen editor. To facilitate this a strict format is used; an example of which is shown here.

If the program finds any deviation from the format it will give an error message and display the offending line of data, and allow the user to edit the file in an on-screen editor window. An example of which is shown in the adjacent screenshot. The puzzle will also be rejected if it has more than one solution.

## Checking for Errors

Clicking on the hand button causes the program to check the puzzle for errors (which can occur due to the player removing incorrect candidates). If errors are detected the hand will flip to the thumbs down position. A further click on the hand will reverse all moves made by the player, back to the last time the puzzle was error free. Note that when a hint is requested the program always checks the puzzle for errors first. If errors are detected no hint search will be performed, the hand will flip to thumbs down and the squares with errors will be highlighted.

## Difficulty

Analysis of all the possible combinations of pairs of the operators +, -, x and / with the digits 1 to 9 shows that there are 5292 different sums. Of these, 8 are satisfied by only one set of numbers, 98 by 2 sets, 70 by 4 sets, ..., 3 by 48 sets. For example A / B x C = 9 is only satisfied by 6 / 2 x 3 = 9; and A + B x C = 119 is satisfied by 8, 9, 7 and 9, 8, 7; but A + B + C = 16 is satisfied by 48 different sets of numbers.

Obviously a major determinant in the difficulty of solving these puzzles is the number of different sets of numbers which can satisfy each individual sum. The next factor is the particular ways in which the sums intersect: each number is in two sums and hence must satisfy both. Finally, the numbers 1 to 9 must each appear exactly once, and so, as for Sudoku, this can further help to reduce the number of alternatives.

To allow the player to set the difficulty of the puzzles presented the Puzzle menu contains an option to select how many sets of numbers satisfy each sum. This value is the average number of sets over the whole puzzle.