Reserve Combat Units for the Mathematically Challenged

Factual Questions
1

(Hi. I promise there is a question at the end of this. It’s starting to look like one of those word problems I just wanted to skip way back when I took my SATs, but I know you guys can knock this off without blinking.)

On November 29, 1943, American forces comprised primarily of U.S. Army units invaded Makin Atoll in the Gilbert Islands. I can’t remember exactly how many troops were theoretically committed to the invasion, but it was a huge number, especially compared to the 800 total Japanese defenders. The 27th Division, tasked with the invasion, numbered approximately 16,000 troops in toto.

It turned out to be a nasty little fight, with green American troops firing blindly into the night ineffectively at harassing Japanese troops. In fact, there was a perception at the time that the whole operation was rather ineffective–it certainly got the attention of Marine Gen. H.M. “Howlin’ Mad” Smith. According to one story, military historian S.L.A. Marshall was called in after the battle to determine what, exactly, went wrong and how it could be corrected.

One of Marshall’s observations is probably repeated in every military history class taught in America. At the time, the U.S. Army worked on a roughly “triangular” system: three platoons to a company, three companies to a battalion, three battalions to a regiment, three regiments to a division. (In WWII, the US used regiments instead of brigades, as noted in this thread.)

Marshall discovered that every company kept one platoon in reserve, every battalion kept one company in reserve, and every regiment kept one battalion in reserve, and one regiment was kept as the divisional reserve. The result was that a very significant percentage of combat troops were in fact in reserve, and not attacking.

Okay. So I’ve stated the word problem. What is the mathematical formula for determining exactly how many troops were kept out of combat? If someone knows an easy way of estimating such a thing, I’d love to know that, too.

There is more to this story, but it’s not relevant to this already bloated question. Thanks in advance for your help.

2

Forces held ‘in reserve’ are NOT out of combat, at least not in the long-term sense. A reserve is a fundamental military concept, it’s a force you hold uncommitted so they can be used wherever might be necessary (as reinforcements, or a counterattack, whatever) as the battle commences. Of course, this gives you less immediate strength starting the battle, but often times the flexibility to respond to later events is worth it.

As for how much were used as that at any given time, well that depends largely on the commanders involved and the situation they’re involved in. The formation and use of a reserve is a matter of much debate, and opinions differ. There’s no easy formula that would be accurate.

However, as your quoted article says, its generally safe to assume that any given commander -will- have a reserve, and that reserve will be one of whatever the smallest cohesive group under his command is (for a company commander, that’s a platoon, for a platoon, that’s a squad, etc)

So 33% (under that organization) is a safe guess. In quadrangle setups (four units to a group) then it’d be 25%

3

I may have made an error below, but the general idea is sound (how’s that for covering my ass?):

Let N_p be the number of men in a platoon. I don’t know what this number is, but I presume you do, so that shouldn’t be a problem. Let N_c be the number of men in a company, let N_b be the number of men in a battalion, let N_r be the number of men in a regiment, and let N_d be the number of men in a division.

Since one of the regiments is kept in reserve, we have N_r men not fighting so far. Each of the two remaining regiments keeps a battalion in reserve, so that brings us to N_r + 2 N_b men not fighting.

Each of those two regiments contains two battalions which remain in the fighting, for a total of four battalions. Each of these four battalions keeps a company in reserve, so we have N_r + 2 N_b + 4 N_c men not fighting.

Each of those four battalions contains two companies which remain in the fighting, for a total of eight companies. Each of these eight companies keeps a platoon in reserve, so we have N_r + 2 N_b + 4 N_c + 8 N_p men not fighting.

Notice that the coefficients are powers of two. This is not a coincidence.

Now, N_r = 3 N_b = 9 N_c = 27 N_p, and so N_b = 9 N_p, and N_c = 3 N_p, so N_r + 2 N_b + 4 N_c + 8 N_p = 27 N_p + 18 N_p + 12 N_p + 8 N_p = 65 N_p men total are kept out of action. Since there are 81 N_p men in the entire division, this means that only 16 N_p men out of the entire divisions participate in the fighting, or a little less than 20%.

4

There would in fact be more “in reserve” than suggested by Mekhazzio.

Lets take it from the top.

The Division has three regiments, one of which is kept in reserve. Thats 1/3 out straight away.

Of the two regiments left, each keeps a battalion in reserve. Each battalion is (1/3 of 1/3), or 1/9 of the original total. So the two battalions left out constitute another 2/9 of the original total.

So we have four battalions left in total - two from each of the regiments that was not kept in reserve. Each battalion leaves out one company. Each company is (1/3 of 1/3 of 1/3), or 1/27 of the original total. That takes care of another 4/27.

Now, there are 8 companies left, each with three platoons. Each platoon is (1/3 of 1/3 of 1/3 of 1/3) of the original total, or 1/81. So if each of the eight companies leaves out a platoon, this gives us another 8/81 reduced from the total.

So the original total has been reduced by:

1/3 + 2/9 + 4/27 + 8/81

This adds up to 65/81, or 80.247% of the original total.

So the division actually sent less than 20% of its total fighting force directly into battle. If the total force was 16,000 as you suggested, that means that fewer than 3,200 went stright to the front lines.

The formula for a “triangle” situation like this would be:

1/3 + 2(1/3x1/3) + 4(1/3x1/3x1/3) + 8(1/3x1/3x1/3x1/3)…

and so on. A better mathematician than me can no doubt come up with a general formula to apply to situations like this, but give me a break, i’m a history major.

5

A general formula would be: SUM[2[sup]n-1[/sup]/3[sup]n[/sup]], where n ranges from 1 to however many levels of organization you’re talking about (4, in this case). Each term of this series has the form (3[sup]n[/sup]-2[sup]n[/sup])/3[sup]n[/sup] = 1- (2/3)[sup]n[/sup], so for the case of four levels, the number in reserve is 1- (2/3)[sup]4[/sup] = 80.247%, as you say. It should be obvious that, as the number of levels increases, the percentage of reserve increases, approaching 100%, so, had there been additional levels or organization above the divisional level, or below the platoon level, an even greater number of troops would have been in reserve.

6

I would add that, as the fighting goes on, the percentage of men who have been committed at some point goes up very quickly. Even if a regimental commander never commits his reserve battalion as a whole, a badly mauled company will be yanked out of line and put into the reserve battalion, and a fresh company from the reserve battalion put in its place. So while total percentage of men committed at any one time remains roughly the same, a larger percentage of the total see combat at one time or another.

7

Beautiful. Thanks everyone for laying down the theory for me, as well as some of the practicalities.

Among other things, I think a variant of this formula might be (loosely) applicable to the perceived parity in numbers which Union troops complained bitterly about in the Civil War. Union armies often had one and sometimes two levels of organization above that of Confederate armies (and usually outnumbered the Confederates by a lot), and the Confederates often did not keep small-unit reserves.