#22 - Local

[Administrator]

Please post below with any questions!

[jaclyn.s]

Can someone please help me understand why C is wrong and E is correct?

This was my thought process for getting C....

R: G Y Z --> G O --> G O
S: F L --> F L --> L K M
T: K M O --> K M Y Z --> F Y Z

--> means one trade

In the first trade, O2 exchanged for Y3 Z3.
In the second trade, F1 exchanged for K2 M2

Am I missing another way for Trustcorp to have no class 2 buildings?

[James Finch]

Hi Jaclyn,

The best way to look at this game is to abstract the individual buildings out to what I'll call "point values," with the class 3 being worth 1 point, class 2 worth 2 points, and class 1 worth 4 points. Each group starts and must always be at 6 total points. At the beginning, we have:

R: 4,1,1
S: 4,2
T: 2,2,2

Here, T is going to get rid of all 2 point buildings, which means the only other possibility for it is to add up to 6 points is to have one 4 point building and both 1 point buildings. As a Must be True question, then, the correct answer choice will be one of or both the 1 point buildings, since we know both of those must be owned by T, but either 4 point could be, as they're equal in all but name. (E) is the only answer choice that indicates a 1 point/Class 3 building, making it correct.

Hope this helps!

[jaclyn.s]

Assigning point values in this game is not intuitive for me and I found that supplementing the class names with numbers was too much for me to track it in this already confusing and difficult game. So instead, I just carefully did one trade at a time by shifting around the variables. I just read the set up for this game. Would you be able to explain how to best get to the answer with the numerical distributions? Also, can you explain how the numerical distributions were deduced?

[Adam Tyson]

It's definitely not intuitive, jaclyn.s, but is instead very deliberate. We often coach our students to think about numbers in every logic game. How many variables, how many spaces, can there be ties or repeats, what's the biggest group and what's the smallest groups, etc. In this game, there is a very clear numbers issue related to possible trades. A Class 3 building can only be traded either for another Class 3, or else the two Class 3 buildings must stay together and be traded for a single Class 2. Same for the Class 2 buildings - they can only swap for another Class 2, or as a pair for a Class 1.

So what does that mean numerically? It means the two Class 3 buildings, which start out owned by the same company, will never be separated. The only way to split them up is trade one of them away for another Class 3, but that will never happen! So, the first big inference of the game, even without assigning values to the buildings, is that Y and Z will always move as a unit.

Another good inference is that the Class 2 buildings are all entirely interchangeable. They all have equal value, so anyone who has one of them can swap with someone to get a different one. Any combination of 3 Class 2 buildings is therefore possible. Also, anyone who has three Class 2 buildings at any point could either swap one of them for the two Class 3s, so it's always possible that someone could end up with four buildings - two Class 2 buildings and two Class 3 buildings. Or, they could swap two Class 2 buildings for a Class 1 building and end up with a Class 1 and a Class 2.

At some point we should recognize that within each class, there is nothing to differentiate one building from another. The rules for F are exactly the same as the rules for G, for example. Anything that is true of one of them is equally true of the other. They are fungible, to use a term you'll be using in your Contracts classes before too long. Same goes for the 4 Class 2 buildings, and for the two Class 3 buildings (although we now know they will always be together.)

So, if you aren't going to assign values, you have to at least think of the possible combinations that could result from some number of swaps. There's no point thinking of the specific buildings at this point because within each class there is no difference. K, L, M, and O are all identical. F and G are identical. Who cares which one we are talking about? Focus on the numbers, which means the Classes and/or the relative values.

Here are some possible distributions, then:

R starts off with a 1 and the two 3s. That means he could swap a 1 for another 1, or a 1 for two 2s, or those two 3s for a 2. The resulting possible combinations, then, after one swap, are:

1-3-3
2-2-3-3
1-2

S starts with a 1-2 combination. After a single swap, he could swap a 1 for another 1, or a 2 for another 2, or a 1 for two 2s, or a 2 for two 3s. Our possible combinations are starting to repeat!

1-2
2-2-2
1-3-3

Finally, T starts off with 2-2-2. He can swap any of those 2s for another 2, or could swap two of them for a 1, or could swap one of them for two 3s. The possible results, after one swap, are:

2-2-2
1-2
2-2-3-3

That's it! We keep coming up with the same four possible combinations, which are:

1-2
1-3-3
2-2-2
2-2-3-3

Because swaps can keep happening forever, anyone can eventually end up with any of these combinations, but the combination held by one company will impact what the others have. For example, if someone - anyone - ends up with a 2-2-3-3 combo, then the 3's are all used up, and nobody can have a 1-3-3 combo. It also means two of the 2s are gone, so nobody can have a 2-2-2 combo. As a result, the other two companies MUST each have a 1-2 combo! Count up the buildings and you'll see that we have accounted for two 1s, four 2s, and two 3s. Voila!

Notice that there is no way to get two 1s in the same combination, no matter how many times you swap. Every combination leads to one of these other combinations, and the individual combinations lead to specific combinations of combinations. This is a doozy, a real brain-buster, if you don't play with the numbers. Assigning values can help, but it's not the only way to do it, as long as you are comfortable with permutations and combinations, old friends from probably middle school math.

If you are going to stick with using the letters assigned to the buildings, you will almost certainly waste a lot of time and put in a lot more effort than this game deserves or requires. Notice that the buildings are completely interchangeable within classes, and then focus solely on the classes. List off to the side which buildings are which class, and then you can use the numbers to create a bunch of different letter combos. The 2-2-3-3, for example, could be any two of the Class 2 buildings with Y and Z. Six different solutions match that. The 1-2 combo could be solved 8 different ways.

Back to where we started, and you're right, this is NOT intuitive. You have to make a conscious decision to play around with the numbers here, one way or another, because that is what this game is all about. Not buildings - Classes. I didn't see it at first, but when I did, I decided that Class 3 buildings were each worth $1.00. That meant that Class 2s were worth $2.00 and Class 1s were worth $4.00, and then I added things up and saw that everyone started off with $6.00. Since every trade is always for equal value, no matter how often you swap you will always have your original $6.00 value in place! That's why nobody can have two Class 1 buildings - that's $8.00, not possible!

That was a lot, I know, and the odds of us seeing a game quite like this one again are slim, but the lessons learned here about the crucial importance of numbers is worthwhile. Give a listen to Episode 31 of our podcast, which breaks down the truly awesome September 2019 games section, and you'll hear a LOT of talk about numbers and their importance!

[Adam Tyson]

And after all that, I lost sight of the original question! How do we answer #22, where T owns no Class 2 buildings? Check out the possible combos that include no Class 2s, and there is only one choice: a 1-3-3. That means T has to own either F or G, but we don't know which, and T must also own both Y and Z, the two Class 3 buildings. That's what proves answer E!

[jaclyn.s]

Thank you so much, Adam. Your explanation helped out a lot. I appreciate it