# The Book

*Professor Povey's Perplexing Problems* is a personal book which celebrates a passion I have always had for playful problems in physics and maths. Questions that rely on material no harder than that studied in high school, but which encourage original thinking, and stretch us in new and interesting directions.

This book is a collection of some of my favourite pre-university problems in physics and maths. These questions are devised to encourage curiosity and playfulness, and many are of the standard expected in some university entrance tests. These questions are perplexing, puzzling, but—most of all—fun. You should regard them like *toys*. Pick up the one that most appeals, and play with it. When you have exhausted it, you can entertain a friend with it. It might seem impossibly nerdy, but for me almost nothing is as enjoyable as being baffled by an apparently simple problem in an area of classical physics I *thought* I understood. This book is a way of sharing the pleasure I have taken in some of the physics and maths puzzles I have most enjoyed. The questions will appeal to those high-school students who have mastered the basics, and feel they have room to play a little with something more unusual. Teachers can also use the problems to stretch their pupils with something more challenging or unconventional. Here I am grateful to those teachers who allowed their classes to pilot these problems, and to the students who bravely attempted them.

I invented many of the questions myself. Others were suggested or inspired by friends and colleagues with an interest in this book. Still more are well-known classics. Even working outside a syllabus however, it is hard to be entirely original with questions in physics and maths. Although these questions can be asked in essentially unlimited ways, the number of fundamental concepts underpinning them is relatively small. Even the ones I am most proud of will no doubt have been asked in numerous other forms over the years.

There was one problem I was unable to solve. Many of these questions—especially the harder ones—work best as *discussion problems*. They are not meant to be asked and answered straight. Instead they should be discussed as part of a tutorial-style conversation, with an experienced tutor helping the student along, giving hints, correcting mistakes, providing encouragement, and so on. There is no real substitute for this kind of help. A question which might seem impenetrable can suddenly yield when a tutor provides just the right hint. I would encourage you to try the problems with friends, or subject tutors, or anyone with a sound knowledge of physics and maths.

I should say a few words about the level of difficulty of the problems. Every single one of them is challenging in some way. The publisher asked me to use stars to indicate their relative difficulty. Although I think this is quite subjective, I had to concede that it would be a shame for someone to happen to try all the most difficult problems first and feel defeated. So I have indicated how hard *I* found the problems the first time I tried them. If you think I have rated some of them too generously, it simply means your intuition is better than mine in that particular area. In general, those questions with one or two stars should yield to a student who is on top of the pre-university high-school syllabus. The following descriptive gradings should give you some idea of what to expect.

★ Not too difficult. Requires thought, and some insight, but should be soluble without hints.

★ ★ Challenging. Requires considerable thought, and some insight, and may require simple hints.

★ ★ ★ Hard. Requires significant thought, and considerable insight, and more substantial hints. In some cases, questions with three stars are only suitable as discussion problems.

I have also included a handful of exceptionally hard questions, to give the most confident students an opportunity for a real challenge.

★ ★ ★ ★ Exceptionally hard. Suitable for discussion. May be too difficult for most students to solve without substantial help. Complex and unusual, these questions allow for different and non-standard approaches

This book has been what I call my “Saturday project”, and there are, I increasingly discover, always fewer Saturdays than I hope there will be. I have not had the luxury of someone to work with, so there will certainly be mistakes, and errors of logic. They are all my fault. But I hope they are few enough that this book brings more enlightenment than confusion. The most challenging aspect of putting it together was striking the right balance with the difficulty of questions. The questions range from being quite straightforward to really quite hard—they are as unpredictable as many university entrance tests. Here I am indebted to the many teachers, students and colleagues who were kind enough to provide detailed comments on the difficulty of the questions. I would be delighted to receive further constructive notes and any genuinely unusual questions which could improve a future edition of this book, and thank in advance anyone willing to provide this feedback. You can contact me by emailing tom@tompovey.com

I hope you will find friends you can share the fun of these questions with, and argue the case for improved or more elegant solutions (many of which exist). To share your solutions with a wider audience, and to learn from the ideas of others, you can contribute to the Puzzle Forum at

## Daglish G. July 21, 2016

Possible alternative soln. to problem 2.8 - 3 door (flap) problem.

Consider a probability space for this problem. 0 - for goat, x - for gold. 3 choices for flap; 3 possible arrangements of content of flap array.

0 0 x 1/3

0 x 0 1/3 3 possible arrangements behind flaps

x 0 0 1/3

1/3 1/3 1/3 Unity.

3 flaps

Probability of each arrangement - 1/3. Probability of going for a given initial flap - 1/3. Each cell - 1/9.

Probability of drawing a goat (i.e. a goat will appear in any flap) - 2/9.

Probability of NOT drawing a goat. - 7/9.

Probability of getting gold on first attempt - 1/9.

Probability of betterment is (7/9 -1 /9 = 2/9 = 2/3).

The probability of discovering gold on the second flap opening is 2/3.

## Daglish G. July 21, 2016

Perhaps I should have said that once the Man had got a goat in his raising of the flap that his apriori probability then froze the probability for NOT getting a goat to an aposteriori 7/9.

## Daglish G. July 21, 2016

Further, in the initial choosing of the flat by the contestant, the psychological probability or expectation stood at 1/9. However the objective probability is frozen a 7/9 on the opening of the flap which reveals a goat. This is the probability that the remaining flap and the chosen one have a probability of 7/9 for not having a goat. Therefore the subjective probability for opening a second flap to show gold has increased by (7/9 - 1/9). This may be taken as an indication for action by the player.

## Daglish G. July 21, 2016

Also, referring to the foregoing, if we choose a flap "X" and define the arrangements: A(x,0,0); B(0,x,0); C(0,0,x), then the probabilities of A or B being in place at our choice of flap "X" is 2/3. For a showing of a goat to any choice of a flap it needs either of two arrangements to be in place at the choice of a flap. The probability that we opened "X" in the first place is 1/3. The probability of the Events E("X",AorB) or E("X",AorC) or E("X",BorC) is 2/9.

## Daglish G. July 22, 2016

Errata: "Probability of Betterment" line in 1st comment should read:

Probability of betterment is (7/9-1/9) = 6/9 = 2/3.