Simultaneous Equations Sudoku

Solve the equations to identify the clue numbers in this Sudoku problem.

Age

11 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

by Henry Kwok

Rules of Equation Sudoku

Like the standard sudoku, this sudoku variant has two basic rules:

Each column, each row and each box (3x3 subgrid) must have the numbers 1 through 9.
No column, row or box can have two squares with the same number.
The puzzle can be solved by finding the values of the 9 given variables in the squares of the 9x9 grid. At the bottom and right side of the 9x9 grid are numbers, each of which is the sum of a column or row of variables. Altogether a set of 16 equations can be formed from the columns and rows of variables and constants.

For example, in the first column beginning from the left of the 9x9 grid, we can form the following equation:

$$c + g + k = 17$$

In the fourth row beginning from the top of the 9x9 grid, the following equation can be formed:

$$k + g + m + c = 23$$

If you are not sure where to begin, you can reveal a hint by clicking the button below:

You might like to start by considering the $4^{\text {th}}$ row and $4^{\text {th}}$ column.

After solving all the equations, the puzzle is solved by the usual sudoku technique and strategy.

Some people approached this problem by first solving the simultaneous equations and then filling in the sudoku. Caleb, Megan and Tamsyn from Frederick Irwin Anglican School in Australia filled in the sudoku grid with letters and then solved the simultaneous equations to replace the letters with numbers. This is Megan and Tamsyn's work (click on the image to see a larger version):

Caleb, Megan and Tamsyn didn't show their working for the simultaneous equations, but they correctly completed their sudoku grids.

Saanvi from Newcastle under Lyme Prep School in England, Neel from Zurich International school, Darren from Harrow School in the UK, Abhinav from St. Olaves Grammar School in England, Ci Hui Minh Ngoc Ong from Kelvin Grove State College (Brisbane) in Australia and Jiali showed how they solved the simultaneous equations.

Neel, Abhinav and Darren started from the fourth column and fourth row. This is Neel's work:

Saanvi and Ci Hui Minh Ngoc Ong started from the first column and fourth row. This is Saavni's work:

Since the difference between $c+g+k$ and $k+g+m+c$ is $m,$ $m$ is $23-17$ which is $6.$

In the third row up, $g$ must be $7$ as $13-6=7.$

In the fifth row, $p$ must be $4$ since $11-7=4.$

In the sixth column, $f$ is $9$ since $22-13=9.$

In the second row, $e$ must be $1$ because $10-9=1.$

In the sixth row, $h$ is $5$ since $14-9=5.$

In the first row, $c$ is $8$ because $19-11=8.$

In the first column, $k$ must be $2$ as $17-15=2.$

By process of elimination, $a$ must be $3,$ but with this information you could easily work that out.

Jiali started from the sixth column and the seventh row. Jiali also suggested how to then fill in the sudoku:

1st Step: Find the values of the variables in the sudoku.

Since $g + m = 13$ and $g + m + f = 22,$ therefore $13 + f = 22$ and $f = 9.$

Since $f = 9$ and $f + h = 14,$ therefore $h = 5.$

Since $f = 9$ and $f + e = 10,$ therefore $e = 1.$

Since $e = 1$ and $p + e = 5,$ therefore $p = 4.$

Since $p = 4$ and $g + p = 11,$ therefore $g = 7.$

Since $f = 9, g = 7$ and $f + g + a = 19,$ therefore $a = 3.$

Since $g = 7$ and $g + m = 13,$ therefore $m = 6.$

Since $m = 6, h = 5$ and $c + m + h = 19,$ therefore $c = 8.$

Since $c = 8, g = 7$ and $c + g + k = 17,$ therefore $k = 2.$

2nd Step: Solve the sudoku.

After finding out what the variables represent, draw up a grid and write the numbers (that you have found) in the place of the variables. This should make it easier to solve the sudoku. Now there's the usual sudoku that you can solve. I solved mine by filling out all the 6's first, then gradually worked down the possibilities of where the numbers can go.

Indii A, Saskia B and Mignione K from Frederick Irwin Anglican School and Darren sent in partially completed sudokus to show their journeys to the correct solution. Here is Indii A, Saskia B and Mignione K's work, which shows a few wrong turns.

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For younger learners

Simultaneous Equations Sudoku

Rules of Equation Sudoku