Colin Singleton, Author at New �Ӱ�ԭ��

Enigma 871 : Pandigitals

Colin Singleton — Fri, 19 Apr 1996 23:00:00 +0000

UNCLE JOE had written the digits 0 to 9 on ten cards, some red, the others
blue, one digit per card. “Now boys,” he said, “you have to arrange the red
cards on the table to form a number, and divide the blue cards into two groups
to form two separate numbers. The `red’ number must be the product of the `blue’
�Գܳ��.”

“Like this?” said Tom. “Or this?” said Dick. “Or this?” said Harry.

“All different, and all correct!” replied Uncle Joe.

Only Tom had included a single-digit number in his arrangement. What was
Harry’s `red’ number?

A £10 book token will be awarded to the sender of the first correct
answer opened on Thursday 23 May. The Editor’s decision is final. Please send
entries to Enigma 871 New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London
SE99 0BB. The winner of Enigma 865 was Ronald Key of Bangor, Gwynedd.

Answer to Enigma 865

Infinite fours

Alan’s number is two.

Correction to Enigma 870: See if you can get the answers if Nora moves to
Education. Ed.

ENIGMA: Infinite fours

Colin Singleton — Sat, 09 Mar 1996 00:00:00 +0000

A GROUP of friends stand in line, each holding a number written on a
card.

Alan’s number is the square root of one more than the number that Brian
holds.

Brian’s number is the square root of four more than Colin’s number.

Colin’s number is the square root of 16 more than Dave’s number.

Dave’s number is the square root of 64 more than Eric’s number.

This is virtual reality, where even the impossible is easy. The line is
infinitely long, and the constant 1,4,16,64 … increases by a factor of four
at each step.

What is Alan’s number?

A £10 book token will be awarded to the sender of the first correct
answer opened on Thursday 11 April. The Editor’s decision is final. Please
send entries to Enigma 865 New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street,
London SE99 0BB. The winner of Enigma 859 was Martin Reddy of Edinburgh.

Answer to Enigma 859

Oh Lucky Day!

16th January 1983

ENIGMA 862: Distances

Colin Singleton — Sat, 17 Feb 1996 00:00:00 +0000

THE MILEAGE chart shows the distances between various secret government establishments, each of which is designated by a three-digit code. The distance between two locations can be calculated by adding the differences between the three pairs of corresponding digits, ignoring the signs of the differences. Thus the distance between 689 and 773 is (7−6)+(8−7)+(9−3)=8.

For reasons of security the government wants these locations to be as far apart as possible, and is concerned that two of them are only four miles apart. Locations 000 and 999 must be retained, but the other four can be moved to locations represented by any three-digit codes, to make the closest pair as far apart as possible.

What is the greatest possible distance between the closest pair?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 21 March. The Editor’s decision is final. Please send entries to Enigma 862, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 856 was Antionette Harris of Stockton, Cleveland.

Answer to Enigma 856

Royal birthdays

Eric

Crunchingly big numbers

Colin Singleton — Sat, 10 Feb 1996 00:00:00 +0000

“WHAT number comes after infinity?” ask children struggling to comprehend the infinite. In our worldly lives we can never see the coastline of our infinite mathematical island – still less glimpse the sea beyond. But Clifford Pickover’s Keys to Infinity unlock the gates to some of the coast-bound highways and byways, and invites us to admire the scenery. We visit the Valley of the Sea Horses, familiar to aficionados of the Mandelbrot Set, and Mount Fractilia, a lesser-known cousin. Pickover lists many of these code-names so that they may be studied and reproduced.

If we tire of one route, we can turn the page, to explore another conceptual corner. You can fill your computer screen with the Golden Curlicue (a deformed Christmas tree), or one of its infinite family. These are fractals – shapes whose detail appears similar at any level of magnification. Pickover’s program generates a forest of Christmas trees, mine generates only one, ever growing, tree. One of us, I suspect, is suffering from the dreaded compounded rounding error disease.

The book poses many questions in its 31 chapters, and gives the reader the keys to develop the answers. Many short programs appear to help. Some of the coding is a little suspect, and the text contains a few silly mistakes. These ate minor obstacles because the book, rather than presenting definitive solutions, encourages us to improve our efforts and even go one better than Pickover.

Not all the investigations require a computer. Chapter 1 asks what proportion of numbers contain the digit 3? (Virtually all of them if you think about it.) Later, we are asked to find a 10-digit number whose n-th digit (n = 0 to 9) indicates the number of occurrences of the digit n in the number. Other chapters are more philosophical.

Pickover’s favourite topic is the squares of a chessboard which seem far from infinite, but larger versions may overwhelm your computer. And my curiosity was bitten by the “Vampires” – pairs of equal-length numbers whose products contain the same digits as the numbers, for example, 1827 9846 = 17988642 – and my thirst whetted by “Fractal Froth”.

Keys to Infinity

Clifford A. Pickover

John Wiley & Sons

Enigma: 852 Sheep Pens

Colin Singleton — Sat, 09 Dec 1995 00:00:00 +0000

OLD MacDonald had a farm – and on his farm he had some sheep pens. Five rectangular pens, in fact, one being completely surrounded by the other four, as shown in the sketch, which is not to scale. The pens are formed by 12 straight lengths of fence, all different lengths and each a whole number of yards long (ignore the size of the corner posts). What is the smallest total length of fencing which can form such an arrangement?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 11 January 1996. The Editor’s decision is final. Please send entries to Enigma 852, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winenr of Enigma 846 was Chris Finn of Beverley, East Yorks.

Answer to Enigma 846

Winners on the left

(1) No; (2) Yes.

ENIGMA: 849 Bird watching

Colin Singleton — Sat, 18 Nov 1995 00:00:00 +0000

ON Bird Island the straight North-South road from Puffin Rock to Quail Moor passes Nuthatch Wood and Osprey Lake. These four locations divide the road into three sections, each a whole number of miles in length.

Rail Marsh is due East of Nuthatch Wood, and its straight-line distances from the four locations on the road are all whole numbers of miles.

The official bird reserve is a triangle with vertices at Puffin Rock, Quail Moor and Rail Marsh. Osprey Lake is equidistant from these three locations – a distance in miles which, we are told, is less than the number of bird species resident on the island.

Using the above information, the new warden, who had just counted the species, was able to calculate the area of the reserve (in square miles). What is it?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday, 21 December. The Editor’s decision is final. Please send entries to Enigma 849, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 843 was Michael Wickins of Studley, Warwickshire.

Answer to Enigma 843

How many are whole?

FORTUNE = 3701284

Enigma: No 836 Who buys the drinks?

Colin Singleton — Fri, 18 Aug 1995 23:00:00 +0000

A GROUP of friends were in the Rose and Crown, debating who should pay for the round. By chance they were seated round the table in the sequence Alan, Brian, Charlie, David … with alphabetically consecutive first initials up to Mr Smith’s.

They took part of a pack of cards, shuffled it, and placed it face down on the table. They agreed to draw one card each, Alan, then Brian, and so on round and round the table, until someone drew a black card. That man would buy the drinks. If they had studied the cards first, they would have discovered that they had more than half the pack and that they had equal chances of drawing the first black.

In the event, the drawing process lasted as long as it possibly could with that selection of cards. What was the initial of the man who bought the drinks?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 21 September. The Editor’s decision is final. Please send entries to Enigma 836, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 830 was Ian Billing of the Hague, The Netherlands.

Answer to Enigma 830

Post Mix

Brian Andrews, Belle View, Altrincham.

Enigma: No. 826 Crescents

Colin Singleton — Fri, 09 Jun 1995 23:00:00 +0000

A RECTANGLE has sides &tgr; and 1/&tgr;, where &tgr; is the golden ratio &tgr; = ½ (√5+1). The circle circumscribing the rectangle is drawn, and semicircles are constructed on each of the four sides.

What is the total area of the four “crescent moons”, within the semi-circles, but outside the large circle?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 13 July. The Editors’ decision is final. Please send entries to Enigma 826, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 820 was Andrew Plant of Lymington in Hampshire.

Answer to Enigma 820

Hard cheese

No, you cannot clear the 6 squares in 1000 moves.

Enigma: Breeding like …?

Colin Singleton — Fri, 14 Apr 1995 23:00:00 +0000

THE rabbits on the island of Hermaphra each produce a litter of offspring each year, and become more prolific with age. The colony originated in the year dot, with just one newborn individual. The following year she/he/it produced just one offspring. In the year dot +2 the original colonnist produced two offspring, and the previous year’s newborn produced one. In each subsequent year, each individual gave birth to a litter equal in number to her age in years.

In which year did the population reach one billion?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 18 May. The Editor’s decision is final. Please send entries to Enigma 818, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 812 was Andrew Plant of Lymingham.

Answer to Enigma 812

Upon my word! 27

Enigma

Colin Singleton — Sat, 07 Jan 1995 00:00:00 +0000

Deductions

Alan has told Bill and Charlie that he had chosen two different integers, each in the range 1-15, and has noted their product and their sum. He shows Bill the product, and Bill remarks that he cannot deduce the pair of numbers.

Alan then shows Charlie the sum, and Charlie, who has heard Bill’s remark, but not seen the product, remarks that he cannot deduce the pair of numbers.

Bill then declares that he can now deduce the two numbers, but does not name them.

Charlie then names the two numbers.

What are they?

A £10 book token will be awarded to the sender of the first correct answer opened on Thursday 9 February. The Editor’s decision is final. Please send entries to Enigma 804, New �Ӱ�ԭ��, King’s Reach Tower, Stamford Street, London SE99 0BB. The winner of Enigma 798 was A. M. Smith of Chelmsford, Essex.

Answer to Enigma 798

Noughts and crosses

aO bO cX

dX e f

gO h i