The Mathematics of 6 on 5 Offenses

Richard Hunkler, Ph.D.
Slippery Rock University
03/01/05

Have you ever wondered why the places on the 4 - 2, 6 on 5 offense are numbered the way they are? The players on the front line, the line nearest the goal, from the offensive teams' left to right are numbered 1, 2, 3, and 6. The players on the back line, the line further from the goal, are numbered 4 and 5. Left and right are defined as if the offensive team members are facing the goal at which they are shooting. Now why aren't the players on the front line numbered 1, 2, 3, and 4, and those on the back line numbered 5 and 6? I told you this is going to be about math - look at the number of numbers that have already been used in the first paragraph of this article.

Well, is it because the geometry of the 6 on 5 offenses is non-Euclidean? I don't think so. I think it is numbered this way because of the history of water polo. I have been told by coaches and administrators that when exclusion fouls were added to the rules, teams used the 3 - 3, 6 on 5 offense almost exclusively. In the 3 - 3, 6 on 5 offense the players on the front line are numbered 1, 2, and 3, and those on the back line are numbered 4, 5, and 6.

The 4 - 2, 6 on 5 offense was a transformation, another good mathematical term, of the 3 - 3, 6 on 5 offense. You math buffs know what is coming next, and that is the transformation was created by rotating the player numbered 6 on the back line to the front line. As the number 6 player rotates up to the number 3 player's position on the front line, the players numbered 1, 2, and 3 shift to their left, and the 4 and 5 on the back line shift to their right to create a 4 - 2, 6 on 5 offense. Thus, the players on the front line are numbered 1, 2, 3 and 6 and those on the back line they are numbered 4 and 5. Eureka! Eureka! I assure you I am not going to run naked down the street of Alexandria because of this discovery as Archimedes, an excellent mathematician, did when he discovered the principle of buoyancy.

Just when you thought you were finished with basic math, I am going to present you with some more geometry. But first we have to review the philosophy of how to score with the 6 on 5 offenses. Math, geometry, philosophy - does a person get credit for a college course after reading this article? Let's discuss the 4 - 2, 6 on 5 offense first. Understanding that a water polo ball can be passed laterally 10 to 20 times faster than a goalkeeper can eggbeat laterally, it doesn't take a mathematician to figure out what you need to do to score on the 4 - 2, 6 on 5 offense. Move the ball until the goalkeeper is committed to one side of the goal, and then pass it quickly to a player on the other side of the goal for a quick shot or move the ball until the defenders guarding the 2 or 3 players are late getting in to position so either the 2 or the 3 player can receive a quick pass for the close-in shot. Use Figure 1 to visualize the triangles I am about to discuss.

 __________
 |                    |

O1        O2            O3        O6


O4            O5


Figure 1

Some of the basic patterns for passing the ball to accomplish the above tactics are called long-triangles and short-triangles. There is both a left and right long-triangle and a left and right short-triangle. The left long-triangle is created using the 1, 4, and 6 players. Since these three players are not on a straight line, their passes outline a triangle. The 1 player can throw the ball to either the 4 player or the 6 player. If the 4 player receives the ball the 4 player can throw it back to the 1 player or to the 6 player. If the 6 player is passed the ball then this player can pass to either the 1 or 4 player. This is continued to the goalkeeper over commits to one side of the goal and a fast pass is made to a player on the other side for a quick shot, or if because of all the passes and the defenders shifting from player to player the 2 or 3 are left open, a hard pass to one of these players is made and a quick shot is taken. The right long-triangle is formed with passes among the 1, 5, and 6 players. Again you are trying to have the goalkeeper commit to one side of the goal so a fast pass can be made to a player on the other side for a quick shot on goal.

The key phrases are "fast passes" and "quick shot" because slow passes will prevent the goalkeeper from having to commit to a side of the goal, and a slow shot allows the goalkeeper to move to and block the ball. When the players are passing then one or two pumps or fakes to the goal are recommended; however, when the goal keeper is caught to close to one side of the goal the offense player on the other side from the goalkeeper is to catch and shoot the ball quickly - no pumps! One or two pumps from the shooter allow the goalkeeper to move to the ball, and reduce the shooter's chances to score.

The left short-triangle is made using the 1 and 4 players. Wait just a time out; two players can form a straight line segment but not a triangle! Remember in the term "student athlete" the word "student" comes first, so where is the triangle? Once the goalkeeper is isolated to the left side of the goal, the 1 player passes the ball to either the 5 or 6 player for a quick shot or the 4 player passes the ball to the 6 player for a quick shot. The last pass to the shooter forms one of the following triangles: the 4 - 1 - 6 or the 4 - 1 - 5, or the 1 - 4 - 6. Note that the 4 player is not allowed to throw the ball to the 5 player for a shot because this is a very error-prone pass and a pass that that allows the goalkeeper to rebound a short distance to the position of the shooter.

The right short-triangle is created with the 5 and 6 players. After the goalkeeper commits to the right side of the goal the 5 player is to pass to the 1 player for the shot, and the 6 player is to pass to either the 1 or the 4 player for the shot. Again the 5 player is not allowed to pass to the 4 player for a shot on goal because of the same reasons explained in the above paragraph.

It is now time to take up the 3 - 3, 6 on 5 offense. Does this mean we can leave geometry and triangles to mathematics classes? If you want to decrease your chances of scoring with the 3 - 3, 6 on 5 offense you can, but if you don't want to do this then you better pay attention to the use of a triangle in the next discussion. Setting up the 3 - 3, 6 on 5 offense as depicted in Figure 2 is going to give you more grief than a math pop quiz.

 __________
|                     |

O1        O2         O3


  O4         O5         O6


Figure 2

Why is this poor positioning of your players? Because there are only two defenders to guard the three players on the back line and if you keep the three offensive players in a straight line it is much easier for the two defenders to guard them. Again the triangle comes to the rescue of the 6 on 5 offenses, and it is an isosceles triangle at that. The kind of triangle in which the length of the 4 - 5 side and the length of the 5 - 6 side are always equal. Figure 3 displays the correct way to position the players on the back line.

 __________
|                     |

O1        O2         O3


  O4                      O6

  05


Figure 3

Now, the two defenders trying to guard the 4, 5, and 6 players will have a more difficult time of doing it. Forming an isosceles triangle almost assures that only two of the players in the isosceles triangle will be guarded, so one player is always free to receive the ball and to make the shot on goal. As the 4, 5, and 6 players are passing the ball, they are to move slowly forward for the best shot on goal. Remember as the 4, 5, and 6 players move forward they must maintain the isosceles triangle to keep one of the players free to receive a pass and to take the shot.

And you thought that mathematics was an intellectual exercise that should be only confined to the chalky black boards of a stuffy classroom. Dude your wrong! Mathematics is an important tool not just for the esoteric but for the practical as well, and more importantly, mathematics can make you a much better water polo player. This is true because water polo similar to mathematics is all about creating simple solutions to complex problems.

Email Coach Hunkler at rhunkler@waterpoloplanet.com