December 6, 2022

Cartesian Battleship


The Battleship game has been very popular since it’s creation in 1931 by the Milton Bradley Company. Players develop a sense of using coordinate systems by calling out coordinates that represent “shots” at the opponent’s naval ships and by responding with the words “hit” and “miss” as shots are called out against their own grid. With a few simple modifications the game may be modified for a more rigorous treatment of the Cartesian coordinate system, and students will actually ask to do their homework! In this article I will describe the changes that are desired, and I will provide everything needed on my website. Modifying the store-bought game has a definite cool factor, but alternatives or given for proceeding without purchasing or modifying a commercial version.

Cartesian Battleship
Dr. Robert F. Szalapski

Figure 1.

Using a template downloaded from the game Battleship becomes Cartesian Battleship.

§ 1. The Goal

The goal is for students to learn through play. The Cartesian coordinate system is of extreme importance in mathematics ever since it was introduced by René Descartes in the 17{}^{\mathrm{th}} century. It is used for graphing and coordinate geometry all through high school and beyond.

§ 1.1. Deficiencies of the Traditional Game

The deficiencies of the of the Milton Bradley version are equally well virtues depending upon one’s perspective. The game has been simplified to a 10\times 10 grid. The columns are numbered 1 through 10, and the rows are specified by the letters A through J. The cell in the upper left corner is A-1, and the lower right corner is J-10. The labels are large and clear, and most elementary-school students are able to play with about five minutes of instruction in the form of guided play. Upon completing a couple of games children may focus more on strategy and less on guessing. It’s a very simple game with a lot of appeal.

However, if we want to teach the Cartesian coordinate system, we need some enhancements. The most obvious is that letters must be replaced by numbers. Coordinates should be called out as ordered pairs of numbers such as (2,3). And, the coordinates should be represented as points at the intersections of grid lines rather than open cells. This is a bit more complicated until students learn that the horizontal coordinate precedes the vertical coordinate.

Next, the coordinate grid requires a proper origin which is point (0,0). For a single-quadrant grid with only positive numbers the origin should appear at the lower left corner. The store-bought game represents the fourth coordinate replacing the numbers -1 through -10 along the vertical axis with the letters A through J. This was an unfortunate choice since it would be more natural to choose the first quadrant when excluding negative integers.

§ 1.2. Enhancements in Cartesian Battleship

At my website,, you may download the template to convert Battleship into Cartesian Battleship. It is located in the Games directory under the Mathematics section for Teachers. Instructions for modifying your Battleship game are included in a Section 2. Here I discuss the merits. One version of the modified game board is shown in Figure 2, and a second version, with enhanced colors to distinguish positive and negative coordinates, is shown in Figure 3. One board or the alternative may be more appropriate for different groups of students.

Figure 2.

A Battleship grid with Cartesian coordinates.

Notice the origin with coordinates (0,0) near the center of the game board. All four quadrants are present. Because the array of peg holes is 10\times 10 the origin cannot be located at the exact center of the board, so a choice was made. Horizontal coordinates run from -4 to +5 while vertical coordinates run from -5 to +4. Grid lines are clearly displayed, and coordinate pairs represent the points at the intersection of the grid lines.

Relatively small labels have been placed along the coordinate axes as would be done on any graph. However, larger and more easily read coordinates have been placed around the outside of the figure where they cover the original labels on the game boards.

Figure 3.

A Battleship grid with Cartesian coordinates with different colors for positive and negative coordinates.

§ 2. Transforming Battleship

Go to the Cartesian Battleship section at and download the template of your choice. Initially it will look like Figure 4.

Figure 4.

Cartesian Battleship as downloaded.

As discussed in Section 3, this version may be laminated and utilized with dry-erase markers. This section is devoted to the more cool version that modifies the store-bought game. We still want to laminate the template, but then use a rotary cutting tool or an exacto knife to trim the board to what you see in Figure 2 or Figure 3. Warning! When printing the template, make certain that page scaling is set to “None”. Check that the dimensions of the blue border of the playing area comprise a square with a 14.5\,\mathrm{cm} edge. Getting the scaling perfect was a challenge to creating the template. A heavy and rigid paper stock is recommended.

Punching the holes is a challenge. I used a screw punch which has a hollow-core drill bit. The most easily obtained screw punch I could find is marketed by Martha Stewart for about $25. The local stores which stock them in the crafting sections were all sold out, so I ordered one on-line. There are 100 holes to be punched in each of four cards to transform a single two-player game. The results are shown in Figure 5.

Figure 5.

A screw punch and a fully punched board.

What remains is to secure the boards into the game. I carefully taped them in place with a strong but clear tape. I cut thin strips of packing tape for the job. The clear board may be carefully removed for the taping if desired. I positioned ships and pegs to hold the new board in place while taping so that the new holes and original holes are properly aligned. See Figure 6 and Figure 7.

Figure 6.

The “enemy” grid with modifications.

Figure 7.

The “defend” grid with modifications.

§ 3. Cartesian Battleship Light

Suppose you don’t have the store-bought version. Maybe you don’t have a screw punch. Perhaps you want many sets and your don’t have the time to punch thousands of holes. Students may use dry-erase markers with laminated game boards on heavy paper stock, or markers can be used with printed sheets that will go the recycle bin after a single use. Each student will need two game boards. One will display the student’s own ships, and the other represents the “enemy”. A cardboard folder may be used to hide the board with the ships from the view of the opponent.

When preparing to play each student will need to place the ships on his or her own card by circling a line of consecutive grid points for each ship. The length of each ship is specified in Table 1.

Ship Number of Grid Points
Aircraft Carrier 5
Battleship 4
Destroyer 3
Submarine 3
Patrol Boat 2

Table 1.

Each player has five ships. Each ship spans the number of grid points specified in the table.

Players should use one color such as black or green to outline their ships. Red may then be used to mark hits, and blue may be used to mark misses. See, for example, Figure 8.

Figure 8.

Using dry-erase markers with laminated cards as opposed to modifying a store-bought set. Use a folder to hide the “attack” card from the opponent’s view.

§ 4. Observations

I developed this modified game to work with a particular group of students in a particular classroom while student teaching for Mr. Ed Perdue at The Norman Howard School. It has been working well for those students, and they enjoy playing the game despite initially making errors with negative versus positive integers and the ordering of horizontal versus vertical coordinates. However, the game boards were left in plain view during Homeroom and Advisement periods. Students were intrigued by Dr. Rob’s enhancements and have been eager to play when time is available. I have watched middle-school students learn Cartesian coordinates for the first time. They are intrinsically motivated to learn, especially when paired with an older student to play the game. I suspect they would be less motivated by the worksheet approach!

§ 5. Graphics with FeynByMe

In my days as an academic physicist I authored a graphics tool, FeynByMe , which generated PostScript Feynman diagrams for inclusion in papers and presentations. I used that to generate the templates for Cartesian Battleship. I then used LaTeX to convert the PostScript images to properly scaled PDF format and to create JPG or PNG images for the Web. Eventually I plan to release FeynByMe for general use.


Free Download: Cartesian Battleship Template

This version has uniform colors for coordinate lines.


Free Download: Cartesian Battleship Template

This version has colors to distinguish positive and negative integers.


Free Download: Cartesian Battleship Article

Download a PDF version of this web page.


Free Download: Smart Notebook Introduction

Download the interactive Smart Notebook file to introduce students to the concepts and to practice ordered pairs as a group. The grids used in the notebook are identical to the game grids.


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