A Leap of Logic 2 |

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.

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.

**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.

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.

**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.

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

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.

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