## Friday, July 4, 2014

### What does it take to change an Army?

Here is a new great puzzle my coworker shared with me. By the way, he is in a delivery room right now waiting for his twin boys to emerge. All the best, Jonathan!

In a faraway land, the Queen's Army consists of 3 Corps.

Each Corp is composed of 3 Divisions.
Each Division of 3 Brigades.
Each Brigade of 3 Regiments.
Each Regiment of 3 Companies.
And finally each Company of 3 Soldiers.
Soldiers are either Air Force soldiers or Marines

Now, the rules, from the bottom of the hierarchy to the top:
Each Company is composed of either:
2 Air Force soldiers and 1 Marine (in which case it is named The Air Force Company)
or
2 Marines and 1 Air Force soldier (in which case it is named The Marines Company)

Each Regiment is composed of either:

2 Air Force Companies and 1 Marine Company (in which case it is named The Air Force Regiment)
or
2 Marine Companies and 1 Air Force Company (in which case it is named The Marines Regiment)

and so on till:

Each Army is composed of either:

2 Air Force Corps and 1 Marines Corp (in which case it is named The Air Force Army)
or
2 Marines Corps and 1 Air Force Corp (in which case it is named The Marines Army)

At the present time the Queen's Army is Marines Army. However Queen's new lover is an Air Force pilot and he convinced the Queen to change the Army to the Air Force Army. The question is: what is the minimum number of soldiers that the Queen needs to change so that her Army will become The Air Force Army?
The answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon

One. Changing a single soldier from Marine to Air Force will change the company he is in, which in turn will change the regiment, and so on until the entire army is likewise changed to Air Force.

Anonymous said...

It seems there are 3 to the 6th or 729 soldiers in all. The puzzle very rigorously states that the proportions remain strictly 2/3 and 1/3 whether of marines or air force. So the difference between an Army of one or the other will always be 1/3 of 729 or 243.

Margaret and Fiona

Jerome said...

This puzzle and the one last week are really super. Who ever comes up with them is to be congratulated.

I think without formal proof that it is one.

Before doing that however let us figure out what the army consists of.
If she had only 1 company, that would equal 3 men
If she had 3 companies, she would have 1 regiment and 9 men
If she had 3 regiments, she would have 1 brigade and 27 men
====
Each time she adds another classification she would have to multiply by three to get to the next level.
Since there are nine different classes, she has 3^6 people in her army. That's 729 men in total.

Now here is the beginning of our analysis.
Starting with a regiment and its 3 companies this is how the 9 men are divided.
M = Marine
A = Air Force

2 M 1 A
2 M 1 A
1 M 2 A
Totals = 5 marines and 4 Air Force = 9

We should try this again with a Brigade
2 M 1 A ... 2M 1A ... 1M 2A
2 M 1 A ... 2M 1A ... 1M 2A
1 M 2 A ... 1M 2A ... 2M 1A

Total M = 14 A = 13

In General the men divide up as
M = (3^n + 1) / 2
A = (3^n - 1) / 2

==================
So now we have to deal with the answer
Start with 9 men. See the actual break up in the discussion of a regiment to see that we get 5 Marines and 4 in the Air Force. We have to cheat a little bit with the definition.

2M 1A
2M 1A
1M 2A

Now if you exchange 1A on the top row with 1 of the Marines in the second row, what you get is ...
3M
1M 2A
1M 2A

Now you have 3Marines which makes it a marine company and 2 Air Force companies. End result the army, if this was its total, would make up an air force regiment.

I believe that it works the same way all the way up. I don't know of another way of doing this without cheating.

Unknown said...

Total number of soldiers equals 3^6=729. Out of them, 365 are marines & 364 are from air force(!!!). To make the army to an Air Force army, just reduce one marine and add one airforce soldier. That is, only one marine to be replaced by a Air Forca man. (Nice problem).

Unknown said...

Total number of soldiers equals 3^6=729. Out of them, 365 are marines & 364 are from air force(!!!). To make the army to an Air Force army, just reduce one marine and add one airforce soldier. That is, only one marine to be replaced by a Air Forca man. (Nice problem).

Anonymous said...

Converting Queen's Marines Army to Air Force Army:
->Marines Army comprises of 2 Marines Corp and 1 Air Force Corp. If 1 of the Marine Corp is converted to Air force Corp then the Army would have 2 Air Force Corp and 1 Marine Corp making it an Air Force Army.
->Marine Corp comprises of 2 Marines Divisions and 1 Air Force Division. Here, 1 of the Marine Division has to be converted to Air Force Division so that we would have an Air Force Corp.
->Marine Division comprises of 2 Marines Brigades and 1 Air Force Brigade. Here again, 1 Marine Brigade needs to be converted to Air Force Brigade so that we would end up with an Air Force Division.
->Marine Brigade comprises of 2 Marines Regiments and 1 Air Force Regiment. 1 Marine Regiment needs to be converted to Air Force Regiment so that the Brigade would become an Air Force Brigade.
->Marine Regiment comprises of 2 Marines Companies and 1 Air Force Company. 1 Marine Company needs to be converted here so that we would have an Air Force Regiment.
->Marine Company comprises of 2 Marine and 1 Air Force soldier. Here, 1 Marine has to be replaced by an Air Force Soldier so that we would have an Air Force Company and eventually an Air Force Army.

Just 1 Soldier!

Lulu

Maria said...

Just one soldier. Adam, Jerome, Baskar and Lulu got it right.
Margaret and Fiona - I will give you a welcoming puzzle point for bravery:)

To change an Army we need to change the Corps composing it from 2-Marines, 1-Air Force to 2-Air Force, 1-Marines. So, we need to switch one Corp from Marines to Air Force.

To change one Corp, we need to change one Division from Marines to Air Force.
To change one Division, we need to change one Brigade.
To change one Brigade we need to change one Regimen.
To change one Regimen we need to change one Company.
And finally, to change one Company from Marines Company to Air Force Company we need to change one Soldier in it from Marine to Air Force.
Just one!