## Introduction

Computational Thinking has become a popular and apparently respectable concept, particularly in education, but I am not at all comfortable with it. It is a rebranding of a selection of mental tools (thought processes), and the danger is that enthusiasm for Computational Thinking will result in pupils being restricted in their thinking because they lack the tools that happen not to be in that selection. My personal view is that Computational Thinking is just a subset of mathematical thinking, but we do not need to argue about that. The point I want to make is that it does not cover all the tools one needs for computer coding, let alone for real life. Generations of social forces and poor education have made "Mathematics" a poisonous brand for pupils at school and for people who are highly educated in non-numerate disciplines. So I can see that inventing a new brand might seem a good idea. However, the descriptions I have found of Computational Thinking omit important parts of the mathematical thinking toolkit and that makes concentration on Computational Thinking dangerous.

Here are some examples of an old family of puzzles you may have met before. I want to remind you of these - or introduce them if they are not familiar to you - because they provide an example where understanding requires the use of tools that are missing from the Computational Thinking kit - in this case, the concepts of symmetries and invariants - two sides of the same mathematical coin.

If it were just a matter of puzzles then this would not be important. However, there are real-world practical problems whose abstract models are very like these puzzles, though more complicated. If, say, you accepted the task of coding an algorithm to help with one of those problems then you would probably start by making an abstract model similar to these puzzles and code your algorithm to solve that. Computational Thinking would certainly be necessary, but would not be sufficient. If you didn't use tools that are missing from Computational Thinking then the success of your algorithm would be a matter of luck.

A well-rounded syllabus ought also to include other, less mathematical modes of thought. We need people to grow up with a better understanding of Scientific Thinking, Logical Thinking, Statistical and Probabilistic Thinking, Ethical Thinking, Thinking about Risks, Skeptical Thinking, Historical Thinking and so on. Overselling "Computational Thinking" certainly threatens the rest of mathematical thinking, but it looks likely to threaten these other capabilities as well.

## Puzzle board

The puzzle question is always: "can you cover all the blank squares with marker tiles?" Each tile covers two adjacent squares and cannot overlap another tile. This first board is not much of a puzzle; Please use it to get to know how this implementation of the puzzles works:

• Click or tap a blank square to add a marker tile
• Click a marker tile to rotate it
• Click the same marked square several times to rotate the tile through its valid positions and then remove it

Do please have a good go at each puzzle before going on to the next.

If after trying puzzles 1 to 6 you haven't understood then you can go on to a page that gives you a hint.

If that hint is not enough then you can ask for more clues. Those clues are given one at a time. To get each one you have to answer a question. Feel free to use your favorite search engine or other reference materials, but well done if you can get the answer without.

Trying to answer is enough to get you the question for the next clue, but getting the answer right will give you more help.

By the way, where the answer is a person, the answer is usually their surname. The answer, being a name starts with a capital letter.

If your answer is not the one I wanted then you will see a scrambled version of the answer. Try again.

## Puzzle No 1

### Can you cover all the blank squares with marker tiles?

As you see two squares have been blocked off leaving you to try covering the rest.

Is this any harder than the completely blank board? Why?

## Puzzle No 2

### Can you cover all the blank squares with marker tiles?

As you see two squares have been blocked off leaving you to try covering the rest.

Is this any harder than the completely blank board or than Puzzle No 1? Why?

## Puzzle No 3

### Can you cover all the blank squares with marker tiles?

Again two squares have been blocked off leaving you to try covering the rest.

Is this any harder than the previous puzzle? Why?

## Puzzle No 4

### Can you cover all the blank squares with marker tiles?

A different two squares have been blocked off this time. Some people find this puzzle easier to understand than No 3

How do you find this compares with puzzles 1, 2 and 3? Please explain?

## Puzzle No 5

### Can you cover all the blank squares with marker tiles?

There are more squares blocked off this time, but the aim is the same.

How do you find this compares with puzzles 1 to 4?

## Puzzle No 6

### Can you cover all the blank squares with marker tiles?

Here is another with more squares blocked off.

How do you find this compares with puzzle 5? Please explain?

If you haven't got it then try the Hint and see if that helps.

## A Mathematical Hint

### Can you cover all the blank squares with marker tiles?

Here, by way of a hint is a puzzle where the board has been reduced.

This puzzle has a mathematical explanation. When you have worked it out, look back at the previous puzzles and find their explanation too.

The explanations are not the same, but they involve the same sort of reasoning.