When playing with numbers it is almost certain that you will discover properties and patterns that you didn't expect at the beginning.

Numbers have inspired people to think of games that rely on their properties. Resolf is such a game. Properties of numbers and patterns are the building block of Resolf. Let's see how the game works.

The principle of the puzzle is simple and can best be explained by an number, in this case the SumPuzzle. The same concept applies for the other variants, as we will see later.

The aim of the puzzle (see picture below) is to put the given 6 play values X = (0, 3, 5,6, 8, 9) into the 6 nodes of the play board such that the sum of the surrounding node values equals the given field values (18, 19, 17).

The solution is shown in the picture on the left and the notation is: (9, 6, 5, 3, 8, 0). The labeling is as follows, from top left to the right and then down to the next row. The equivalent system of equations for the solution is:

18 = 9+6+3

19 = 6+5+8

17 = 6+ 3+8+0.

Above we saw the summation variant. There are three more variants of the Resolf puzzle and each of them are matched to a distinct color.

1. ProductPuzzle → red
2. SumProductPuzzle → blue/red (divided in sub-variants)
3. FunctionPuzzle → green.

In the ProductPuzzle variant, which is matched with the color red,  multiplication is the operation to solve the puzzle. The aim of the puzzle is to

``Place the play values/expressions in the nodes, such that the product of  the surrounding node values equals the value/expression in each field."

The next variant is a combination of the sum- and product puzzle, the SumProductPuzzle and therefore matched with the colors blue and red. The aim of the puzzle is to

``Place the play values/expressions in the nodes, such that the sum or product of the surrounding node values equals the value/expression in the corresponding field."

As an exercise try to solve the puzzle above.

As last we come to the FunctionPuzzle, where the equations in the fields are functions and the chips are coordinates. The aim of the puzzle is to

``Place the coordinates in the nodes such that the surrounding coordinates satisfy the equation in each field."

As an exercise try to solve the puzzle above.

The examples we saw above made use of numbers, but Resolf can be played with algebraic objects like formulas with variables as well. Various variants are possible, with logarithms, roots, functions and coordinates. For more examples you can have a look at the end of the article or visit the website of Resolf. On the website you can also download a free app with much more examples!

Resolf provides flexibility to create puzzles that range from easy to more challenging. You can construct puzzles in order to offer exercises of a suitable difficulty that will help develop arithmetic skills.

A game that you can do with two people or two groups is the following.

First you construct each a puzzle, then you exchange the puzzles with each other and solve the puzzle of your neighbor. It is advisable that you first agree about the range of the play values. Below you can find more detailed instructions.

You can also help by giving one or more hint(s), placing a play value in the correct place. 

And in case you need an even bigger challenge you can add some conditions to the puzzle. For instance, given are the play values ( 3, 5, 6,7,8,10). Then the following conditions should also be satisfied:

• The field values $v_1 , v_3$ are equal. That is $v_1=v_3$.
• The sum of the field values is 67. That is $v_1 + v_2 + v_3 = 60$.

The goal in this case is to determine the field values $(v_1, v_2, v_3)$ by placing the play numbers so that the conditions are satisfied.