*By Michael Hartley*

'Le Monde' is one of the biggest newspapers in France. Like all big newspapers, they publish puzzles. In 2013, they published a video explaining a new puzzle, 'La Grille De Sommes' (The Grid Of Sums). If you speak French, you can watch the video here. If not, read on!

The Grid Of Sums starts with an empty grid. Either a 3x3 square grid, or a 4x4 grid or bigger if you feel up to the challenge.

Then, you choose empty grid cells one by one. Each time you choose a cell, you add up its neighbors, and write the total in the cell you chose. If all its neighbors are empty, you write '1'.

Keep going until the grid is full. Your goal? Make the last number you write as large as possible!

Here's an example. I started with an empty 3x3 grid. First of all, I chose a corner, and wrote 1. Then, I chose the opposite corner, and wrote 1 there as well:

Next, I chose the middle square. The neighbors are the two corners I already filled. These add up to 2.

Then I chose the middle cell of the top row. It's only got two filled neighbours, I add them up and get 3.

Next, the other corner of the top row gets filled with the number 2+3=5.

The last empty square on the right has four neighbors! I add them all up, getting 3+5+2+1, or 11. It looks like I'm making progress! I write the number 11 in the cell.

Now, I'll start working back along the bottom row. The neighbors of the middle cell are 2, 11 and 1, which add up to 14.

The last corner cell gets the number 16, since 2+14=16.

Now, there's only one empty cell. It has a lot of neighbors! Adding them up, 1+3+2+14+16, gives 36.

The biggest number I got is 36. That's not a bad score, but it's possible to do better.

I've prepared a whole bunch of grids for you to download and print, so you can try the game yourself now. First of all, there are some square grids - the biggest I've given here is a 7x7 square grid. The numbers will get very large! It's possible to score 150941453 on the 6x6 grid, and 164912612349 on a 7x7 grid. Nobody knows if these are the best possible scores, maybe you can go even higher! The world record for the 8x8 grid is a whopping 639561837115046. That's almost a quadrillion! If you do better than this, let me know.

Here are the square grids:

You can play the Grid Of Sums on any kind of grid. You might like to try it on a hexagonal grid. I've provided hexagonal grids with:

I have no idea what the best possible scores is for each of these grid sizes! You could also try rectangular grids, or look through my collection of unusual graph paper and mark out some wacky grid shapes.
Another variation you could try is this: try to make your biggest number as *small* as possible. My intuition is that this is easier than making your score large - so to make this variation more challenging, you can add this rule: except for the very first move, you can't choose cells with no neighbors.

You could also play Le Monde's Grid of Sums as a two player game.

- Choose who goes first.
- Players take turns choosing a grid cell, and either writing 1 or the sum of the surrounding cells.
- When the grid is filled, each player scores a number of points equal to the highest number they wrote.
- Then, the other player goes first, with a new empty grid.

Have fun!