A Logical Combination

A Leap of Logic 2

In our first post on logic puzzles we looked at some very simple ideas of logic, we learned about inference and deduction and some of the ways they might be applied. We looked common pitfalls in logical thinking and of course we learned how to avoid them. In this post we will begin to explore the different kinds of logic puzzle that can be found, and start to apply the things we have learned.


As Kylie Minogue once sang, you’ve ‘got to be certain’ and in logic puzzles this is the mantra you must keep repeating to yourself. It is true that in some instances you might have to try out a possible solution, to see if it meets the criteria; but for the most part you will be analysing a puzzle and looking for the certainty, the one part of the solution that must be a certain way, lest it break the conditions of the puzzle.

As is becoming habit in Wandering Puzzler, I want to take you through the same puzzle several times, increasing its complexity at each step, in an effort to help you understand how a particular type of logic puzzle works.

Pure Linear Placement Puzzles

A pure linear placement puzzle deals with the placement of a series of items in a line. Each item is unique, and has a single characteristic by which it will be referred. These characteristics will be used only to identify the items, not to compare them.

What do we mean by this last statement, suppose our three items with number cubes showing one, two and three. In a pure linear placement puzzle you might see clues such as

Cube two is not in position two

But you would never see clues such as:

The highest cube is in position three

The second clue is using a comparison between the items to help with placement, and a pure-linear doesn’t have that feature. Let’s get started with the very simplest version of the puzzle we can think of.

You might find it helpful to have some coloured counters, cubes, lego bricks or whatever in various colours, so you can work on the puzzles physically, rather than just mentally. You’ll also find some Wandering Puzzler cube nets on the blog that can be used for a whole variety of up and coming puzzles.

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Logic Puzzles

Two Bricks In A Wall

You have two blocks, one is red and one is blue. Place the blocks in a single straight line, based on the following information. Position 1 can be considered the very left of the line, and the position number increase as you move right. There are no spaces.

A: The Red Block is not in position two.

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Now, that wasn’t hard was it? You should have the Red Block in position one, since Clue A says it can’t be in position two. This means only position two is available for the Blue Block.

The important skill here is that you have identified from the clues, a certainty. Then, with this information in hand you are able to place the rest of the blocks. It might seem simple at this stage, but as the puzzles become more complex, the method will remain largely the same.

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Possibilities

With just two blocks, and two positions, whatever certainty we give in our clues is going to solve the whole puzzle. There were still lots of of options for what the clue might be. To give the same solution I could have said any of the following.

The Red Block is in position one.
The Blue Block is not in position one.
The Blue Block is in position two.
The Red Block is to the left of the Blue Block.
The Blue Block is to the right of the Red Block.
The Red Block is on the left.
The Blue Block is on the right.

As the number of items to place increases the number of possible solutions increases exponentially.

The number of combinations that two items can be placed in is just two. If we add a third item then there are three choices for the item in the first position, and then two ways to arrange the other two. Therefore the number of ways to arrange three items is three multiplied by two; six. 

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Logic Puzzles

Three Bricks in a Wall

You have three blocks, one is red, one is blue and one is green. Place the blocks in a single straight line, based on the following information. Position 1 can be considered the very left of the line, and the position number increases as you move right. There are no spaces.

A: Neither the Green Block, not the Blue Block is in the middle.

B: The Red Block is to the left of the Green Block. 

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This time our certainty, is in fact a certainly not. Clue A tells us that the Green Block does not fall in the middle; or position two. The Blue Block also does not fall in this position; therefore we can deduce a certainty. The Red Block falls in position two.

Clue B tells us that the Red Block is to the left of the Green Block, or of course that the Green Block is the the right of the Red Block. There is only one position available to the right of the Red Block, and this is position three. The Green Block falls in this position.

There is now only one position and one block remaining, so the Blue Block is placed in Position one.

Certainties are the bread and butter of these kinds of puzzles.

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Permutations and Combinations

There were six ways of arranging three blocks, as the number of blocks increases so the number of combinations increases exponentially. Calculating the number of combinations is as simple as working out n!, where n is the number of blocks and that exclamation mark means 'factorial', an easy way of saying multiply the number by every integer lower than it, but greater than one. 

1! = 1

2! = 2 x 1 = 2

3! = 3 x 2 x 1 = 6

4! = 4 x 3 x 2 x 1 = 24

5! = 5 x 4 x 3 x 2 x 1 = 120

10! = 3628800

12! = 479001600

As you can see, the number combinations gets very big, very fast, when you consider that there are 81 blocks to place in a sudoku puzzle, although many of them are duplicates, you realise why working logically is the only way to solve these puzzles. 

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Logic Puzzles

Four Bricks in a Wall

You have four blocks, one is red, one is blue, one is orange and one is green. Place the blocks in a single straight line, based on the following information. Position 1 can be considered the very left of the line, and the position number increases as you move right. There are no spaces.

A: The Red Block is not next to the Orange Block

B: The Green Block not at either of the ends.

C: The Orange Block is in Position 3

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This time our certainty is clue C, which tells us the position of the Orange Block, we then have a negative clue to consider. Clue A tells us that the Red Block cannot be next to the Orange block; which rules our Position 2 and Position 4 so the Red Block must be in Position 1.

We are left with the Blue Block and the Green Block, and Positions 2 and 4; Clue B tells us that the Green Block cannot be at the ends, so that rules out Position 4 for the Green Block so it goes in Position 2.

We have just the Blue Block left which goes neatly into Position 2.

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So you have four blocks under your belt, but we'll be adding several more in the next Wandering Puzzler post, so until then, keep puzzling.