TBTB

Apologies for stasis in the blogspot. I feel guilty. I've been TBTB- too busy to blog. And just a little slow. There's no excuse for this sloth- as the zoo-keeper said when one of his charges misbehaved, so here is at least something.

Where does all the time go? Don't answer that. There isn't time. There's just time for me to break the virtual silence with this tiny puzzle- and just time for you to solve it (sorry I didn't have time to make it any smaller).

It was suggested by a game of noughts and crosses when my opponent failed to turn up and I forgot what the grid looked like. Needless to say, I won that game, though if I'd chosen '0' I'd still be waiting for the game to get started, and even this little kickshaw would not be here.

Hot Hols in Hungarian Hills

I have escaped the vagaries of the English summer and decamped to Budapest for a month or so to finish a book. Here for the last two weeks the temperatures have been in the thirties and high twenties, and so it has been a time for shady walks in the hills of Buda, for lounging about on Margit Sziget and for cruising down the Danube, experiences which make up to some extent for all that rain back in England.

Byron evidently also had a low opinion of weather in his own country as the quotation hidden in the accompanying wordsquare shows.

Find the missing words and insert them in the clue below to find what Byron thought of our seasons. The letters of each component word lie in a straight line vertically, horizontally or diagonally - up or down.

Clue: The (7, 6) - (6) in (4), to (10) in (6).

2-D Disco

There wasn't enough room to give the solution to Pyrgic Puzzle No. 3 on May 7th:

'Down at the 2-D disco, 3 identical discs gang up on a slightly smaller disc and they trap it between themselves and a straight line, as shown.

      If the radius of the small disc is 1 unit, what is the radius of the bigger discs?'

Many readers became lost or bogged down in detail. Solving a problem like this one is rather like swimming from A to B except

that there are no markers to tell you in which direction you must set off and— perhaps as importantly— how far to go. So some readers went around in circles and others gave up before they could reach the conclusion.

You need to find an expression giving R in terms of r. Don't imagine that this must be a linear equation. As it happens it turns out to be a quadratic.

Let's start with the construction shown and let's label as many of these lengths as we can in terms of R and r.

Now AB = 2R; AD = R + r.

Note that the point C is a vertical distance R above the horizontal line. D is r above the horizontal line. So CD = R - r.

BC = R + r - CD = R + r - (R - r) = 2r.

To summarise:

AB = 2R.

AD = R + r.

CD = R - r.

BC = 2r.

There are two ways of finding AC: from the upper right-angled triangle, and from the lower right-angled triangle. Equating these will give us a direct link between R and r.

From the upper triangle:

AC² + BC² = AB², so that AC² + 4r² = 4R², so that:

AC² = 4R² - 4r².

From the lower triangle:
                                            AC² + (R - r)² = (R + r)².

This boils down to:
                                           AC² = 4Rr.

Equating these two expressions for AC², we find:

4R² - 4r² = 4Rr.

This can be written:
                                                    R² - Rr - r² = 0,

and since we are told that r = 1, we have:

                                                 R² - R - 1 = 0,

This may be regarded as a quadratic with R as the unknown.

Most people solve their quadratics by referring to the standard formula for the canonical quadratic, which is expressed in standard form as:

ax² + bx + c = 0.*

This has solutions:

x = [-b ± √(b² - 4ac)]/2a.

In this particular instance, a = 1, b = -1, c = -1, so the solutions are:

R = [1 ± √(5)]/2.

R is positive, so we must reject the negative root and we are left with:

R = [1 + √(5)]/2,

which is known as the golden ratio, which all goes to show that the golden ratio turns up in all sorts of unexpected places.

*If you don't remember or know the formula (if you are short on memory but long on cunning) you can certainly proceed as follows:

R² - R = 1.

Add 1/4 to both sides:

R² - R + 1/4 = 5/4.

The left-hand side is simply (R - 1/2)², so

(R -1/2)² = 5/4.

We can take square roots of both sides without affecting the balance of the equation:

R - 1/2 = ±√5/2

So that:

R = (1 ±√5)2/.

Again we discard the negative root, since it gives us a negative value for R and the Radius of a circle is a positive quantity.

R = (1 + √5)/2

as before.

What 'Pyrgic' means...

Over the last 30 years, since The Pyrgic Puzzler was published and The Guardian started to run what was to become the weekly column Pyrgic Puzzles (now accompanied by its sister column Wordplay), I have often been asked what the word Pyrgic means and how this title came about.

Those with access to a dictionary that is compendious enough have been able to discover for themselves that the word is associated with πύργος or pyrgos, which is the Greek word for a tower, and is found in, for example, Homer; and so it means of, about or suitable for a tower.

 


But how did the name come to be associated with bendy and anti-intuitive puzzles?

It is curious that a book that has so completely transformed my life and which The Evening Standard— perhaps over-generously— claims has changed the face of puzzle-writing, was not written for publication at all, but as a diverting travelling companion for a friend.

An academic acquaintance of mine from Oxford was due to fly to Lesvos to spend two weeks in a tower of the kind that the inhabitants of that island still call a Pyrgos. Now my friend is not a good traveller: no sooner has he arrived somewhere than he wishes himself back home. As the date of departure approached he became more and more anxious.

At that time I was teaching physics and maths remedially to pupils who had failed their A-levels, though to be fair they had first been failed by the educational system. The lessons were gruelling for tutor and pupils alike: there is nothing more laborious than to have to harrow up badly misconstrued ideas and replace them with more logical concepts: it is easier to teach a subject from scratch.

Also nothing that anyone said was to be taken at face value: it had to be established or proved from fundamental principles. Only in this way was it easy to tell with any efficiency whether a pupil had understood something for himself or was merely claiming to do so. It is the basis for making oneself accountable for the knowledge one holds. I set questions which tested their understanding and I stretched their intellectual self-sufficiency with paradoxes.

It was only fair to allow the pupils to get their own back on such a demanding tutor. It also made it clear that all of us are learners and investigators even if some of us have studied longer. Soon they were hanging around after tutorials setting me brainteasers that they hoped would catch me out. If they succeeded it was an occasion for much glee and mirth. How much more pleasure is occasioned by results that have been worked for and earned.

But puzzling in this way does not only amuse; it also stimulates intellectual curiosity: a state of value in any mental endeavour and one that can so easily be crushed in traditional schooling— especially in maths and physics.

Trading in nit-picky puzzles was also a natural way of learning the value of paying close and critical attention to the form of words used in describing situations and ideas, a sloppiness in which is a sadly widespread impediment to clear and logical thinking.

I had jotted down some of the more entertaining of the brainteasers that we set each other, and it was a collection of these that I now pressed into the hands of my acquaintance as he prepared to fly off to his pyrgos. As these puzzles were set in the familiar world of Oxford he only needed to read one of them to be at once transported home, if only in his imagination.

Thus were the Pyrgic Puzzles born and they were duly transported to Lesvos where they proved a success. On my friend's return they found a home on his coffee table in his rooms in Oxford where he taught philosophy. Diligent pupils were occasionally stymied or rewarded with one of them.

And so matters might have rested had Iris Murdoch not dropped by to visit just as my friend was called away to the telephone. When he returned he found the novelist comfortably ensconced on the sofa deeply absorbed in the puzzles. She told my friend that they should be published and proposed she write an introduction.

After the book came out, the question of what I was to do for a career settled itself. There were newspaper columns, consultations, radio programmes and huge numbers of letters to answer from members of the public correcting, informing, enquiring or just quibbling: all about puzzles.

If my friend had not been called away to the phone at that precise moment, Iris Murdoch might not have picked up the puzzles and I might not have embarked on a life which culminated with an Oxford College appointing me its College Enigmatist (a unique post not unlike that of a medieval court jester, but without any of the attendant political risk). Henceforth my work would be play; and my play would be work.

The book was subsequently translated into Italian, Greek, Polish and even American, where it was felt that the over-alliterative title Professor Percival Pinkerton's Perplexing Puzzles would somehow convey more of the spirit of the book to the buyer than the original title, despite the fact that no character called Pinkerton is ever alluded to in the text of the book.

Now the book that started it all is to be reissued this autumn in the United States under its original title by Dover books. It is an honour for The Pyrgic Puzzler to find a new home in the country which gave birth among others to the great puzzlist Sam Loyd, the late-lamented Martin Gardner, and logician Ray Smullyan. I hope the puzzles will amuse and transport American readers just as they once did my academic friend in his tower on Lesvos and the many readers of the weekly column in The Guardian.

                                                                                      Picture by

                                                                            Michael James Harrington

It's High Time - It's Pie Time!

Pyrgic Puzzle 3 (29th January 2011) reads as follows:

One of Ma Rainey's circular blueberry pies fits snugly into a Pie Thagoras™— a baking tray in the form of a right-angled triangle of base 1 unit and hypotenuse 2. The pie is cut with 3 straight-line cuts each joining the centre of the pie to a side of the tray which it meets perpendicularly. A gourmand stands at each apex and takes the piece of pie nearest him. What are their shares?

Who would divide pies in this way? Only in a puzzle! But this puzzle was simpler than some readers assumed. There is, for example, no need to find the radius of the circle, though that is easily done in the case of such a right-angled triangle. How do we know this? It is clear that the fraction you get is independent of the absolute size of the pie.

Nor is there a need to presume a priori that connecting the centre of the circle to the apices of the triangle necessarily bisects the angles of the triangle though of course it does. All that comes out in the wash.

The first step is to realise that the right-angled triangle is half of an equilateral triangle, so that its angles are 30° and 60°.

Join the centre of the circle to the vertices hosting the acute angles.

It is then easy to show that ABC and DBC are congruent triangles (the same triangle flipped over). We know this as they are right-angled, they share a hypotenuse and have the same length for one of the other sides. By Pythagoras, the other side must be the same. As these triangles are congruent, corresponding angles are the same. That means that angle CBD is 15°, and the angle DCB is 75°. So the share of the gourmand at B is twice this, namely 150°. This constitutes 150°/360° = 150/360 = 15/36 = 5/12 of the pie.

The second share taken by the gourmand at the right angle is clearly 1/4.

The third share (by subtraction) must be 1 - 5/12 - 1/4 = 1/3.

This last result can be checked in the same way by considering congruent but flipped triangles at the other acute angle.

Points to Ponder

a) What if the pie tray had angles P, Q and π/2?

b) What if the pie tray were not a right-angled triangle?

For once - four ones!

1111

Not only is 1/1/11 the start of the New Year, but- without the strokes - we have 1111: a palindrome (a number which reads the same if the order of the digits is reversed). It is also a repunit: a whole number none of whose digits is other than 1. The first few are: 1, 11, 111, 1111, 11111 …

I have always thought of such numbers as oners (it helps to name things in the mathematical garden) but since 1966 when Albert H. Beiler coined the term repunit (a portmanteau of REPeated and UNIT) others have used this designation, so must I if I want to communicate and be understood.

There is something aesthetically interesting about numbers exhibiting a high degree of symmetry, so such numbers may seem to some special in non-mathematical ways. To some this manifests itself as a mystical feeling. For example, Uri Geller writes about the significance of the 11.11 phenomenon. Certain numbers seem to crop up significantly more often than others in apparently random contexts.

To many people 11.11 is just a time on the clock, and, worse still, just one among many. We pay more attention to it than to other less striking clock readings- those not having any obvious and immediate symmetrical feel- because it is a special number, not because that time has any special significance (though I do think that since it consists of two elevens it ought to be called ”elevenses”.

I myself, if I ever catch the microwave clock showing 3.14, pause and eat the nearest available approximation to a piece of pie, often saying out loud: ”pi time: it's 'high time for pie' time”. I have regarded this ritual as evidence of playfulness (homo ludens) and an excuse to eat pie rather than anything of cosmic significance. How convenient to be able to blame one's indulgence on some supernatural ukase issued through medium of the microwave. But would anyone believe it?

After all, the appearance of a special number on an object does not make the object special. Nor does it mean that a collection of otherwise arbitrarily-numbered objects that happen to share the same special number are special in their own right.

Besides being patterned, repunits may generate other patterns. For example, 1111 X 1111 = 1234321, a result which some describe as being like a pyramid. This is not, strictly, good as an analogy, as pyramids are 3-D figures. It looks rather like a stepped triangle. In mathematics care is taken to be careful to be accurate with definitions, as a careless use of words and sloppy designations can really lead one up the garden path.

The fact that squaring 1111 produces a stepped triangle is a feature of the squares of all repunits from 1 to 111111111. But thereafter carries become involved in the calculation and the perfection of the pattern is disrupted. (1111111111 X 1111111111 = 1234567900987654321).

Can one use repunits to make patterns that do not break down? Yes. For example, all repunits with an even number of digits may be expressed in the form:

11 = 2 + 3^2
1111 = 22 + 33^2
111111 = 222 + 333^2

and this pattern works not just for the first ten but for arbitrarily long repunits with an even number of digits. This shape is reminiscent of the silhouette of a stepped ziggurat or a sawn of stepped triangle.

Such numerical regularities are interesting and attractive (hence their use in recreational mathematics and puzzles) but their explanation is to be found routinely and quite ”down-to-earthily” in the structural articulation of the multiplication and addition of digits. If we were to watch a mechanical device computing the result of such a calculation we would see the cogs entering regular cycles as a result of the relations between the number of cogs and the digits of the numbers. The eye is good at spotting such patterns, for reasons connected with the evolution of the visual system.

Repunits are of interest not only in recreational maths but also in number theory. In the 19th Century they were studied in relation to repeating decimals, and more recently in relation to the theory of prime numbers.

Points to Ponder

1. An n-digit repunit cannot be prime if n is composite. Why?

2. Show that 1 is the only repunit that is a perfect square.

3. Show that for any odd number you can find a repunit that is a multiple of it.

4. 1/27 = 0.037037037... and 1/37 = 0.027027027... Is this a coincidence?

5. Some mathematicians initially thought that the idea of a repunit was too arbitrary to have any deep mathematical significance as it had too much to do with decimal digits. Were they right?

Cold Turkey

There were no Pyrgic Puzzles in The Guardian last week (25th December, 2010), and some readers spoke of going `cold turkey` this Christmas. But both Pyrgic and Wordplay will reappear next Saturday, so it`s hardly the beginning of the Dark Ages. And besides there are both Fry's English Delight on BBC Radio 4 (December 28th) and my chat with Stephen Fry about Wordplay (see link below) to weigh up. The most that can be needed, then, is a St Bernard bearing a barrel of bafflers; a small Red Cross parcel of brain-food to sustain one, to tide one over this short dark tea-time of the soul. So here`re a few puzzles to mull over in the meantime...

1.”The ICC match referee Ranjan Madugalle said that as captain of his country his actions were unacceptable”.So spake the voice on the evening news on the radio. Pedanticus hit the radio with an old bat. What had bowled him over?

2. Christmas Past saw Scrooge buying just enough crackers for everyone present to pull a cracker with someone. Christmas Present finds him buying just enough crackers for someone to pull with everyone with a single cracker left over. Christmas Future will see him buying just enough crackers for everyone to pull with everyone. Of course no-one would be so frivolous as to pull a cracker with himself, and Scrooge woudln't dream of pulling a cracker with anyone. He always has exactly the same number of guests at all his parties. If the number of crackers he bought for Christmas Past added to the number he bought for Christmas Present equals the number he will buy for Christmas Future how many guests does he always have?

3. Tertius Threeman of Tring had two sons. He left them his field which was in the shape of an equilateral triangle. They were to divide this in two with a single straight-line section of the same sort of fencing as that which already defined the perimeter of the whole plot of land. Tertius clearly had it in mind that each son have the same area, but that's not what the will said, and it is what the will says that ultimately counts. It required only that the fencing used to delineate the perimeter of each of the two shares of land should be equal in length. (Well, it said the two shares must be 'isoperimetric'.) As a result Primus (exercising the rights of his seniority) divided the field in a way that was most advantageous to himself. What fraction was left to Secundus? If originally 6 miles of fencing was required to fence off the land how much extra fencing was required when the land was divided?

4. Each of the 3 Stooges pulls a cracker with each and every other one. Each cracker contains a paper hat. What are the chances that each
one gets a hat? Now do the same, but for the Three Musketeer and d`Artagnan.

5. Mme Ceci (née Cela) bought each of the other 4 members of her family a present each of which she duly wrapped and placed under the Christmas tree and each of which she carefully labelled with a label other than that for the person for whom the present was intended. The four correct labels were there, of course, and the 4 correct presents but each one was mismatched. Of course this is by now a familiar ritual so that no-one picks up a present with his own or her own name on it, but deliberately chooses a different one at random. What are the chances that everyone picking up a parcel with someone else's name on gets his or her present?

6.After only a year's acquaintance an admired but undeserving person was peppered with gifts after the numerical fashion of the old ditty The Twelve Days Of Christmas (On the first day one present, on the second 2 presents plus 1 present, on the third day 3 presents, 2 presents and 1 present and so on with arithmetical inevitability). And when this bout of giving was over the True Love quite naturally asked for feedback, only to learn that the object of his obsessive generosity (if such it was) felt that as they had been going out for a whole year she was still one short. What was the ungrateful and unworthy object of his affections on about?
7. On Uranus the year is longer and so it has been decided that Christmas there has 999 days. How many presents can a Uranian expect over The 999 days of Christmas on Uranus?

8. The initial letters of which carol spell out OSWIM? And which OHAT?

9. What does a selfish person give his friends for Christmas?

comments: pyrgic@puzzlemaster.co.uk

Chris and Stephen Fry discuss Wordplay:

http://bbc.co.uk/programmes/b00wr7rf>