Tag: Puzzles

Figuring: The Joy of Numbers – Shakuntala Devi

On By bookramblings142In ReviewsLeave a comment

As a child I was fascinated by mathematics, but especially by tricks and shortcuts that could be done. I started reading Martin Gardner’s section of Scientific American when I was eleven or twelve years old, I don’t claim to have understood all of it but each month my knowledge of recreational mathematics was stretched just that little bit more. I’ll cover one or more of his books in a later blog. However in 1977, when I was fifteen, this book was published and it was written by somebody who, at least partly, earned her living from amazing feats of mental arithmetic, I had to get a copy, and this book is still on my shelves today. Some of it I already knew but there were whole sections where she explained how to do tricks that I had seen done but which had baffled me such as calculating the day of the week for any date given to you or working out square and cube roots in your head. I remember practising these tricks for hours until I could do them too.

The book starts of simply by looking at each of the digits 0 to 9 in detail, explaining what is special about each of them and giving tips around multiplying and dividing by them, patterns in their multiplication tables etc. She then moves on to chapters about multiplication, addition, division and a very short chapter on subtraction. These chapters not only suggest shortcuts, which I still use today, to perform such calculations but ways to quickly check if the answer you get makes sense such as casting out nines. The book really caught my attention however when we reach calculating squares, cubes, square roots and cube roots. Amazingly cube roots which non mathematicians would assume to be much more difficult then square roots are actually very simple and fifth roots are even easier, square roots proved to be quite tricky. But just to see how easy extracting a cube root lets look at all you need to know, worryingly forty five years later I can still remember this:

  • 1 cubed = 1
  • 2 cubed = 8
  • 3 cubed = 27
  • 4 cubed = 64
  • 5 cubed = 125
  • 6 cubed = 216
  • 7 cubed = 343
  • 8 cubed = 512
  • 9 cubed = 729

Assuming that we are starting with 474,552 (which is 78 x 78 x 78) how do you get the right answer? Well first of all look at the thousands i.e. 474, this comes between 343 and 516 so the first digit is the cube root of the lower number which is 7. Next you will notice that all the cubes in the list above end with a different number and you just need to find the one that ends with the same digit as the number you are trying to extract the root of which in this case is 2 which matches 512 or 8 cubed and there we have the answer, the 7 from the thousands value along with the 8 from the final digit gives the required answer of 78. Notice that it was simply a case of knowing the first nine cubes and no actual calculation was performed on 474,552 in order to get the right answer.

Calculating the day of the week is a bit more tricky as you need to memorise four tables, admittedly the first of which is simply the first four values from the seven times table so this barely counts as a table and the working out is also more involved. I can’t do this in my head anymore and frankly with the all pervading computers or mobile phones with calendars on them what was once a occasionally handy ability is now of no use whatsoever as you are rarely that far from a device where you can look up the day for a specific date if you need it. When I was a teenager however this was quite impressive at least amongst the other maths fans at school and I got to be pretty quick at it.

The book finishes with chapters on special numbers and finally tricks and puzzles most of which, even then, I had already encountered but this book stretched still further my mathematical skills and I loved it. It has been great fun reading it again and finding out what I remembered and what I had forgotten. Shakuntala Devi died at the age of 83 in 2013 and wrote several books on mathematics along with astrology and oddly ‘The World of Homosexuals’ which she claimed was inspired by her marriage to a homosexual man but Figuring: The Joy of Numbers is probably her best known work, at least outside India although sadly it appears to now be out of print. If you know a child interested in mathematics I suggest trying to get a copy for them, it really is a joy.

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Professor Stewart’s Cabinet of Mathematical Curiosities – Ian Stewart

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After a series of novels, time for something factual and an exercise for the brain. Ian Stewart was Professor of Mathematics at The University of Warwick when he wrote this book in 2008 and still holds that title although now Emeritus since he retired. He has written numerous books on mathematics, several of which I own so this was chosen as the first one I picked off the shelf, he was also the third person to write the recreational mathematics column for the periodical Scientific American, taking the reins from 1991 to 2001. This column was started by Martin Gardner back in 1956 and he wrote it until the mid 1980’s and this was the true start of my love of mathematics so it has been a pleasure over the years to have sat in a few bars with Ian and discuss maths and also to enjoy his very readable books.

This book, along with it’s sequels Professor Stewart’s Hoard of Mathematical Treasures’ from 2009 and ‘Professor Stewart’s Casebook of Mathematical Mysteries’ from 2014, are an interesting mix of puzzles and mathematical history and are partly built upon notebooks that Ian started whilst still at school and more snippets that he has gathered over his long career of anything that looked fun or interesting in the field of mathematics. I had come across roughly half of the puzzles before and it’s surprising it was so few as I have lots of maths puzzle books but the 249 pages of puzzles and essays plus 60 pages of solutions and/or further further discussions on points raised contained a lot that was new to me. Of the essays I particularly liked his short summary of Fermat’s Last Theorem and how Andrew Wiles finally came to solve it centuries later. Ian demonstrates his skill as a good teacher in these essays, not simplistic, after all anyone picking this book up will have an interest in mathematics but not too complex either. The solution relies on a whole new branch of mathematics so he doesn’t try to explain how the solution works but instead explains why it is important and hints at the complexity involved. There are also essays on fractals, chaos theory, various famous mathematicians and numerous important conjectures and theorems spread throughout the book.

It is in the puzzles though that Ian allows his wit to shine through, even if sometimes that is just a series of bad puns as in ‘The Shaggy Dog Story’ which is a fun rewriting of a really old puzzle that would be familiar to almost all readers of the book so he dresses it up to still make it fun and then in the solutions section introduces a variant of the puzzle which I hadn’t come across before. The puzzle involves the terms of a will where the eldest son is to have a half of his fathers dogs, the middle son a third and the youngest a ninth. Unfortunately when the father dies he has seventeen dogs so the division looks like it could be quite messy if the will is to be executed exactly. The solution is actually quite easy and I first saw this puzzle over forty years ago but I’d never seen the follow up question which can also be solved where the legacy of the first two sons remains the same but the third son gets a seventh of the dogs and the puzzle is reversed because you have to work out how many dogs the father had in order for there to be a solution with no dogs harmed. If you haven’t seen the original puzzle before I’ll put the answer at the end of this blog.

I’d recommend this book to anyone with an interest in maths, the essays are fascinating, the puzzles fun and you’re guaranteed to learn something new.

I also have both the subsequent books in this style and there is an interesting part to the introduction of the second book, I’ll reproduce it here.

Cabinet was published in 2008, and, as Christmas loomed it began to defy the law of gravity. Or perhaps to obey the law of levity. Anyway, by Boxing Day it had risen to number 16 in a well known national best seller list, and by late January it had peaked at number 6. A mathematics book was sharing company with Stephanie Meyer, Barack Obama, Jamie Oliver and Paul McKenna.

This was, of course, completely impossible, everyone knows that there aren’t that many people interested in mathematics.

Ian therefore unexpectedly received an email from the publisher wanting a sequel which did well, but not as well as the first hence the longer delay before the third book. The Casebook is easily the weakest of the three as too many puzzles are dressed up in cod Sherlock Holmes stories which frankly only serve to pad out the puzzle and it appears to have been remaindered as I didn’t know it existed until planning to write about the first two and got a brand new still shrink wrapped first edition copy for a third the original price seven years after it originally came out.

Dogs problem solution – You just need to borrow a dog from somebody else. This will mean you have 18 dogs, half of that is 9, a third is 6 and a ninth is 2. As 9 + 6 + 2 = 17 you can then give the borrowed dog back, Now try the follow up question…

The Wanderer & other Old-English Poems

On By bookramblings142In Reviews4 Comments

My latest limited edition book from The Folio Society is The Wanderer illustrated and signed by Alan Lee. An artist best known for his decades long association with works by Tolkien, both in illustrating his books and his many years in New Zealand working on the Lord of the Rings and Hobbit trilogies.

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The text is largely from a 1966 Penguin Classic ‘The Earliest English Poems’, translated by Michael Alexander, which also included four pages of Beowulf. Over the years this has been revised until the 2008 edition which provides the entire text for this book, with some amendments, which by then was entitled ‘The First Poems in English’. Lee was approached by The Folio Society to see if he would like to illustrate something for them and between them chose this work as it takes him back to the source materials that so inspired Tolkien in his writings. This is by no means a typical way round, the society would normally choose a book that they wanted to publish and then approach an artist to illustrate it; but what it has produced is a book where you can see the love the artist has for the material and I suspect they eventually had to stop him from creating any more artwork so that the book could actually get published. As it is each poem has its own distinctive decorative borders along with the beautiful tipped in colour paintings and on page printed black and white illustrations.

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The poems and riddles themselves come from a very short window in time, between the reign of King Alfred the Great over the Anglo Saxons (886 to 899AD) where he started the process of moving the written word from Latin to Old-English and the Norman invasion of 1066 when all that was swept away with the imposition of Norman French. In truth there were probably just thirty or forty years where Old-English hit its peak before becoming almost extinct. The greatest source material for the work of this period is The Exeter Book which was regarded as largely worthless for centuries before becoming recognised as the treasure trove that it is.  The poems are much more powerful than might be expected from their great age, they clearly come from an oral tradition as they are directed at the reader as though being read to them, I am reminded of the Icelandic sagas in concept if not in size. Indeed as Bernard O’Donoghue writes in his especially commissioned foreword

There’s a vitality to these poems, written as they were at a time when life was so much more embattled, more desperate and fragile

Along with the general introduction and note on translation each poem has its own introduction setting the scene for the following work and providing mush needed context. The works are over a thousand years old and the people who wrote and read them were very different to ourselves.

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The original Penguin book its variants and companion volumes have sold over a million copies in the fifty years since they came out and the quality of the work shows exactly why Michael Alexander is such a respected translator and this edition makes reading them so much more of a joy than the original paperbacks. The text is presented with the original on the left hand side and the translation on the right as can be seen in one of my favourite works included the fragment of ‘The Battle of Maldon’ from the section of Heroic Poems. I suspect I like these more than the somewhat more introspective other poems is my fondness for the sagas and these have more of a feel of those. However this is an account of a real battle that can be also seen in The Anglo-Saxon Chronicle to such a level of detail that there is also an accompanying map included with the text so the reader can easily see how the fight progresses, which frankly is not well for the English side and a lot better for the attacking Vikings.

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The riddles are great fun and at the back are a set of proposed solutions, however the one that I have shown as an example also has drawings by Alan Lee which somewhat give away the answer. All the riddles are from The Exeter Book where presumably there are a lot more as these start at number seven and there are lots of numeric gaps.

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The answer is of course mead.

As only 750 copies were printed at £395 each and these are all sold out from the Folio Society it would be difficult to get a copy of this fine edition, but if I have whetted your appetite for Old-English poetry and riddles then the Penguin paperback is still in print and considerably cheaper.

There is a short video showing the book from the Folio Society

and a longer video of an interview with Alan Lee.