# Geometric patterns & Gbrainy

The illustration below comes from Gbrainy, a puzzle program distributed with Ubuntu linux.
My wife and I first discovered it on one of our holidays, when our son Piet-Jan had equipped an old laptop with Ubuntu linux. GBrainy gave us several evenings with much needed mental exercise. It features logic, calculation, memory and verbal puzzles and exercises. One of the puzzles was this one:

I must admit it took me some time to figure it out, and you can check the solution.

Some logicians and mathematicians maintain that there are two ways of thinking: deductive and inductive. Deductive reasoning is the type of reasoning where you have all elements available, and your deductions involve logical reasoning, but the reasoning does not include anything new. One might say that your reasoning takes your from general to specific. An easy example: All cows are animals. Bella is a cow. Conclusion: Bella is an animal. Inductive reasoning goes the other way around: from special to general. In the puzzle above this type of reasoning is very clear: you start looking for what the commons properties in the figure are, and what letters can be associated with them. Finding the properties is the inductive part: you don’t know what the common properties are, and you have to discover them.

The puzzle made me thinking, and I designed several others, some of which I took the liberty to present them to the GBrainy group, others are presented here for the first time. Several of my fellow workers, such as Jan Zoomers, Jon Koeter, Michiel Matthijssen and Pieter Vuijk tested some of them, for which I am grateful to them.

(Give up? Don’t check the solution to soon).

(Give up? Don’t check the solution too soon).

# What comes next?

In a previous post I mentioned inductive and deductive reasoning. One might say that in inductive reasoning all elements are known, while deductive logic tries to find the general underlying a pattern.

Famous game inventor Robert Abott designed a game “Eleusis” which is based on inductive reasoning, and though this is a blog on puzzles I hope to spend a future article on this game – there are sufficient similarities.

Another area where inductive reasoning is used, is in many IQ tests. Many of them have exercises consisting of a series of numbers, where the person taking the test is asked to find the next number. Patterns are usually purely based on elementary arithmetic.

Here are some exercises, ranging from elementary to difficult:
1) 3, 6, 9, …
2) 2, 6, 10, 14, …
3) 3, 12, 48, …
4) 19, 15, 11, …
5) 128, 64, 32, …
The sequences above are very simple, one operation is repeated again and again.

Things can however be made slightly more complicated, and this is the level you will find in many IQ tests.
6) 3, 6, 4, 8, 6, 12, 10, … (solution 41)
7) 3, 5, 8, 10, 13, 15, 18, .. (solution 51)
8 ) 3, 8, 3, 11, 3, 14, 3, .. (solution 7)
9) 100, 90, 180, 170, 340, 330, .. (solution 14)
10) 260, 130, 120, 60, 50, … (solution 20)
11) 15, 7, 22, 14, 29, 28, … (solution 28)
12) 18, 36, 13, 18, 8, 9, … (solution 35)
The nice thing about this type of puzzles is that after doing a series, you know the patterns to look for, which makes the next one easier to solve. As a result, you actually score higher when you consecutively take an IQ test where they have this kind of exercises. So yes, solving these puzzles actually makes you score higher at IQ tests.

13) 5, 11, 23, 47, … (solution 58)
14) 38, 22, 14, 10, … (solution 64)
15) 4, 12, 26, 54, … (solution 72)
16) 248, 86, 32, 14, … (solution 79)
17) 3, 4, 7, 11, … (solution 84)
18) 3, 6, 11, 18, … (solution 94)
19) 3, 7, 12, 18, 25, … (solution 104)
20) 3, 5, 9, 15, 25, … (solution 114)
21) 2, 5, 10, 17, 26, … (solution 124)
22) 3, 10, 29, … (solution 133)
23) 4, 8, 24, 96, 480, … (solution 142)

Occasionally, you will encounter sequences which will be posed in a different form, such as:
24) 5, 9, …, 17, …, …, 29. (solution 149)

25) One form occasionally used in puzzle books and magazines is the rectangle or square:

 7 12 5 17 16 18 9 3 ..

26) Puzzle magazines often love to add illustrations. That is nothing new, Sam Loyd in the 19th century added an illustration to almost any puzzle in his Encyclopedia of Puzzles. Here is an example as they might appear in puzzle magazines.
In the supermarket, you find several fruit bags with labels. Unfortunately, one of the labels is damaged. Can you figure out what the price on the damaged label is?

Should the puzzle beat you, you can look up the solution 37

Things can be taken too far. For example, take the next sequence:
1, 2, 4, 8, 16, 31, ….
No, the 31 is correct, it is not a typo for 32, it is really 31.

You may spend days thinking about this sequences and never find an answer. Unless you happen to have stumbled on the problem before, you are not likely to stumble on the solution, which is 57.
The logic behind this series is that it is the number of regions formed by joining points on a circle: