3 Hits to Eliminate a Unit Variant

Introduction

In the original One Hour Wargames rules as written (RAW) a unit is eliminated on 15 hits. In the 3-Hit variant a unit is eliminated on just 3 hits. I originally saw this on John’s Wargame Page. It has become the way I play OHW now.

The 3-Hit variant has some advantages. The RAW version probably requires a fiddly tracking system for the 15 hits such as -

• A paper roster which needs to be marked up. Or

• Use of numeric counters or dice combinations placed against a unit to show hits.

On the other hand with 3-Hit a simple counter can be used. I have glued a red counter to a yellow counter so that -

• On one hit I place a yellow counter.

• On two hits I flip the counter to show red.

• On three hits I remove the unit.

This is quite a simple way to track hits.

The Rules for the 3-Hit Version

So how are hits in 3-Hit determined? In the 3-Hit version you throw 1, 2 or 3 dice depending on what you would have done if using the RAW. Like this -

+------+-------+

| RAW  | 3-Hit |

+------+-------+

| D6+2 |  3D6  |

+------+-------+

| D6   |  2D6  |

+------+-------+

| D6-2 |  1D6  |

+------+-------+

A hit is registered for each 5 or 6 on any dice in the normal unmodified case.

So if you throw 3D6 and the throw is {6, 4, 5} then 2 hits are registered.

If we call a “Positive” modifier, a modifier that gives advantage to the testing unit e.g. Terrain Advantage, and a “Negative” modifier one that gives a disadvantage to the testing unit e.g. Flank or Rear Attacks, then we can summarise the 3-Hit hit rule this way –

                    +--------+

                    | Hit On |

+-------------------+--------+

| Positive Modifier |   +6   |

+-------------------+--------+

| Unmodified        |   +5   |

+-------------------+--------+

| Negative Modifier |   +3   |

+-------------------+--------+

The logic here is simple. A positive modifier that would result in half the diced hits in RAW is replaced with a +6 throw which halves the probability of a hit in 3-Hit. Similarly a negative modifier that would result in double the diced hits in RAW is replaced with a +3 throw which doubles the probability of a hit in 3-Hit.

The Mathematics

So you may ask: Does the 3-Hit version play the same as the RAW version? The answer is “somewhat” and to see this we need to do some mathematics.  If you are not mathematically inclined then skip this bit but the graphs below may be of interest.

Number of Moves to Eliminate a Unit under RAW

Let’s start by looking at the unmodified case. If we throw a single D6 a lot of times and then work out the average throw (= average number of hits) it will be close to 3.5.

We can exactly calculate the average number of hits for the D6 case this way –

Average Number of Hits = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5 hits.

Those of you familiar with statistics will recognise this as the Expected Value.

Since it takes 15 hits to eliminate a unit then -

Average Moves to Eliminate Unit = 15 / 3.5 = 4.29 moves.

For the D6+2 case –

Average Number of Hits = (3 + 4 + 5 + 6 + 7 + 8) / 6 = 5.5 hits.

Average Moves to Eliminate Unit = 15 / 5.5 = 2.73 moves.

For the D6-2 case –

Average Number of Hits = (0 + 0 + 1 + 2 + 3 + 4) / 6 = 1.66 hits.

Average Moves to Eliminate Unit = 15 / 1.66 = 9 moves.

Number of Moves to Eliminate a Unit under 3-Hit

Let’s start with the 2D6 case. If you throw 2D6 there are 36 possible outcomes. Looking at these outcomes we see –

• 4 out of 36 are a combination of 5s and 6s and therefore give 2 hits.

• 16 out of 36 contain a 5 or a 6 combined with a 1-4 value therefore giving 1 hit.

• 16 out of 36 contain combinations of 1-4 with another 1-4 value giving 0 hits.

We can calculate –

Average Number of Hits = (4*2 + 16*1 + 16*0) / 36 = 0.66 hits.

Since it takes 3 hits to eliminate a unit then -

Average Moves to Eliminate Unit = 3 / 0.66 = 4.5 moves.

For the 3D6 case there are 216 possible outcomes –

• 8 out of 216 are all 5s or 6s giving in 3 hits.

• 48 out of 216 are 5s or 6s with one number from 1-4 giving 2 hits.

• 96 out of 216 are a 5 or a 6 with the other 2 numbers being from 1-4 giving 1 hit.

• 64 out of 216 contain neither a 5 nor a 6 giving 0 hits.

We can calculate –

Average Number of Hits = (8*3 + 48*2 + 96*1 + 64*0) / 216 = 1 hit.

Average Moves to Eliminate Unit = 3 / 1 = 3 moves.

For the 1D6 case there are 6 possible outcomes –

• 2 out of 6 are a 5 or a 6 giving in 1 hit.

• 4 out of 6 are neither a 5 nor a 6 giving 0 hits.

We can calculate –

Average Number of Hits = (2*1 + 4*0) / 6 = 0.33 hits.

Average Moves to Eliminate Unit = 3 / 0.33 = 9 moves.

Summary

We can summarise the results thus -

+----------------------+------------------------+

|   Average Moves to   |    Average Moves to    |

| Eliminate Unit (RAW) | Eliminate Unit (3-Hit) |

+-----------+----------+------------+-----------+

|    D6+2   |   2.72   |     3D6    |     3     |

+-----------+----------+------------+-----------+

|    D6     |   4.29   |     2D6    |    4.5    |

+-----------+----------+------------+-----------+

|    D6-2   |     9    |     1D6    |     9     |

+-----------+----------+------------+-----------+

You can see that these numbers are approximately the same so that – on average over many throws – the RAW and 3-Hit versions will give the just about the same number of moves to eliminate a unit. 

So, in conclusion, 3-Hit can replace RAW.

Simulation

To test this out I built a simulation in Excel. All the RAW cases (D6-2,D6, D6+2) and all the 3-Hit cases (1D6, 2D6, 3D6) across 15 moves were run 5000 times each and the results graphed for comparison of the hits profile. (And unlike the fractional Average Moves to Eliminate calculated above in the simulation of course eliminations occur on an integer number of moves.)

The main takeaway here is that the RAW and the 3-Hit have different distributions so that, while they have similar average moves for an elimination, the how and when the hits accrue will be different. Looking at the graphs below you can see -

• The RAW has a more symmetrical and tighter distribution. (In fact in the D6+2 case there is a 68% chance it will take exactly 3 moves to eliminate the enemy unit. You can nearly bank on it!)

• The 3-Hit distribution and this is the interesting part, has a rapid rising distribution, followed by a long tail. So there is always the possibility of a “grand slam” when using 3-Hit. In fact there is 1 in 27 chance that a 3D6 roll will eliminate the enemy on the very first move of combat - when a combination of just 5s or 6s is thrown. The long tail also means that the combat can drag on in a ongoing slug-fest. I happen to think this is a better combat model – either you have an immediate impact on the enemy or the conflict continues for a long while.

November 2023