# How to create a logic grid given three categories?

How do they make the three category setup here? I do not understand how this works. A detailed answer to this is most welcome.

• On your previous question you accepted an answer that explained how this kind of grid is used in solving certain logic puzzles. What exactly about that was not clear? Just saying that you don't understand does not help us explain. – Jaap Scherphuis Apr 7 at 13:35
• What are the parameters taken horizontally and vertically?How do I decided them? How do I prepare the subgrids here? – Rajorshi Koyal Apr 7 at 14:07
• The set of categories are given in the clues of the problem you are trying to solve. The order you place them around the grid does not matter, as long as for every pair of categories there is exactly one block (the three coloured 4x4 blocks in the example - the bottom one presumably should be red as it pairs up the States and Names categories). In practice that means that one category uses the left column of blocks, another category uses the top row of blocks, and all the remaining categories are used twice, once in the rows and once in the columns but in reverse order. – Jaap Scherphuis Apr 7 at 14:46
• What is really meant by a three way setup can you give a practical example.I am really not able to digest. – Rajorshi Koyal Apr 7 at 14:51
• I don't know what else to say. That grid you show in your question is a perfectly clear example with three categories. In that case it is a grid for solving a logic puzzle involving matching up the three categories Names/States/Colours, where each has four options. – Jaap Scherphuis Apr 7 at 15:05

I believe what you are asking is, "how can I go from a set of categories to drawing a grid"? I will demonstrate how to do this in two ways.

### Purpose of the logic-grid set up

First, let us establish the purpose of the logic grid. All categories intersect, but only once. This allows any information relating any categories to be marked down in a specific location. I will demonstrate this on an empty three-category logic grid.

To make talking about the categories easier, I will say that Category 1 is first names, Category 2 is last names, and Category 3 is home country. In this grid, if I say "Lea (first name) does not live in India (country)", I can mark this information down in the intersection of 1 and 3. I would put an X in the cell which is in Lea's row and India's column, because Lea does not live in India.

Information that combines 1 and 2, such as "Lea (first name) has the last name Garcia (last name)" can also be marked down, in the area I labeled 1&2. And finally, information combining 2 and 3 - "Ms. Young (last name) does not live in Korea (country)" - can be marked in the area labeled 2&3.

This is the reason these kinds of logic grids are useful. No matter what information we are given, no matter what combination of categories are in the information, we can put some marking on our grid to represent that information.

If we have information combining more than two categories - "Lea (first name) has the last name Garcia (last name) and does not live in India (country)" - we can break this down into sections that only combine two categories:

• "Lea (first name) has the last name Garcia (last name)"
• "Lea (first name) does not live in India (country)"

Then we deal with each of those pieces of information separately. There are other kinds of clues that are much more complicated, but to put them on the grid still boils down to "Can I relate two categories to each other? Can I break this information down so it only deals with two categories?"

### Way 1 to set up a logic grid: small to big

(please excuse my terrible image editing skills)

To start this method, let us look at a two-category logic grid.

It should be fairly obvious that this is the only way to set up a 2-category logic grid: with one category along the side and one along the top. The order in which we set up the categories does not matter; just that they intersect each other.

How do we get a three-category logic grid from here? A three-category logic grid must have intersections of 1&2, 1&3, and 2&3. We already have the intersection of 1&2. To make an intersection of 1&3, we can simply add a section for 3 next to the section for 2, as in the picture below.

By placing the 3 as another set of columns, we intersect it with the 1 which has rows. But wait! We are not done, because we still need an intersection of 2&3. We make this in a similar fashion as the 1&3 one - place a 3 where it will intersect a 2, in other words, place a set of 3s below 1.

Now all intersections are available and this logic grid is complete. This is how we would build a 3-category logic grid from small to big.

I'll also show how to make a four-category logic grid, to demonstrate how this strategy can be extended.

Looking at the completed three-category logic grid, we see that the only way to make a new 4th category intersect with the 1 category is to make the 4 cover some columns (as the 2 and 3 have columns going down which intersect with the 1). So, we add a 4th category as so:

Okay, now we have intersections for 1&2, 1&3, 1&4, and 2&3. We still need to make 4 intersect with 2 and 3. But how to do that? So far, we have only added intersections with 1 other category at a time.

I will now introduce another rule. Each category can have, at a maximum, 1 row and 1 column placement. This is to prevent a certain intersection from occurring twice. For example, if we placed another 4-column next to the current 4-column, there would be two intersections of 1&4:

But now it's unclear where to put information that involves categories 1 and 4. So this is bad.

Okay, given this rule, and since we have already placed a 4-column area, we must place a single 4-row to intersect with both 2 and 3. Where can that go? How about in between the 1-row and the 3-row?

Take a minute to look at this and understand what just happened, and convince yourself it works. What we did:

1. Put our new category as the farthest left column-group (this is the first thing we did, placing the 4-column next to the 3-column) to intersect with 1
2. Put our new category as the second-from-the-top row-group, to intersect with every other previously-placed category.

### Way 2 to set up a logic grid: memorize the rule

The small-to-big method immediately suggests a rule that we can apply instead of slowly building the grid up. Here is the rule:

1. The column categories, from right to left, are: 2, 3, 4,.... n, where n is the number of categories that must be placed. Note that 1 does not appear as a column.
2. The row categories, from top to bottom, are: 1, n, n-1... 3 (if there is a 3rd category - if there are just two categories, use the simple two-category logic grid), where n is the number of categories that must be placed. Note that 2 does not appear as a row.

This rule works to make sure all the intersections appear for any size logic grid. Let's try it on a 6-category grid! Here is the empty grid:

Using our rule, the columns must be labeled 2, 3, 4, 5, 6, and the rows labeled 1, 6, 5, 4, 3.

Does this have all our intersections? Let's check! We must have:

1&2, 1&3, 1&4, 1&5, 1&6
2&3, 2&4, 2&5, 2&6
3&4, 3&5, 3&6
4&5, 4&6
5&6

Hey look, it worked!

As a final note, when ordering items within a category, here are the rules:

• If this category only appears as a row, or only as a column, then order however you like
• If this category appears as both a row and a column, then the convention is for the top-down ordering (when listing items out for when it is a row) to be the same as the left-right ordering (for listing items out for when it is a column). I'm not sure if anything would actually break if you changed that, but best not to mess with something that works.

As an example of the latter bullet point, in your image "South Dakota" is the leftmost and topmost element of Category 3, the states.

Remember, the assigning of categories to numbers is entirely arbitrary. Just give them numbers and follow one of the systems of setting up the grid, stick to your numbers, and you'll be fine.

• Amazing answer.Can you assist me at times in the chat section? – Rajorshi Koyal Apr 7 at 16:38
• I'd prefer for any questions to be posted to the main site, and I'm not willing to be on-call for chat. Main-site questions can be answered by anyone here and chat is more hidden. – bobble Apr 7 at 16:43