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Programming Enigma Puzzles

30 April 2018

Posted by on **From New Scientist #2220, 8th January 2000**

You play this game by first drawing 20 boxes in a continuous row. You then draw a star in each box in turn, in any order. Each time you draw a star you earn a score equal to the number of stars in the unbroken row [of stars] that includes the one you have just drawn.

Imagine that you have already drawn eleven stars as shown below, and you are deciding where to place the twelfth.

Drawing the next star in box 1 would score only 1 point, in box 11 it would score 2 points. A star in box 2, 5 or 6 would score 3 points, and in box 9, 12 or 19 it would score 4 points. Drawing the star in box 16 would score 6 points.

Your objective is to amass the lowest possible total for the 20 scores earned by drawing the 20 stars.

What is that minimum total?

This puzzle completes the archive of *Enigma* puzzles from 2000. There are now 1169 *Enigma* puzzles available on the site. There is a complete archive from the beginning of 2000 until the end of *Enigma* in December 2013 (14 years), and also from the start of *Enigma* in February 1979 up to January 1988 (10 years), making 24 years worth of puzzles in total. There are 623 *Enigma* puzzles remaining to post (from February 1988 to December 1999 – just under 11 years worth), so I’m about 62% of the way through the entire collection.

[enigma1064]

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5 February 2018

Posted by on **From New Scientist #2232, 1st April 2000** [link]

George has a small field adjacent to his new house, in which he intends to build an ornamental lake. The field is triangular with sides conveniently 4, 7 and 8 times the length — about 12 feet — of the job lot of fence rails he has bought from the local builders’ merchant. He has set the required 19 posts evenly spaced around the perimeter and built the fence.

To define the lake, George has stretched three lengths of rope, each from one corner post to the last post before the corner (working anticlockwise) on the opposite side of the field. The enclosed triangle will be the lake, the surrounding area grassland. The whole field measures two “ares” — an “are” being a metric unit of area equal to 100 square metres.

What is the area of the lake?

[enigma1076]

20 November 2017

Posted by on **From New Scientist #2243, 17th June 2000** [link]

George has made a number of spinners for his children to select numbers when playing board games. Each has a circular disc divided into a number of equal sectors. He has made several discs of various sizes and with various numbers of sectors.

George’s son has discovered that three of the discs will fit together so that the marked radii form a triangle (shaded in the diagram) which includes just one sector on each disc.

Further research revealed that several other sets of three discs of suitable sizes and numbers of sectors can be used for form triangles of various shapes in this way, always including just one sector on each disc. George has found one set in which two of the discs have different prime numbers of sectors.

How many sectors are marked on the third disc of this trio?

[enigma1087]

14 August 2017

Posted by on **From New Scientist #2258, 30th September 2000** [link]

George called a meeting to inaugurate the National Apathy Party, open to anyone who has never voted in a General Election. He hopes to be the next Prime Minister. The turnout was phenomenal, but he managed to seat them all in Wembley Stadium (capacity 80,000). George proposed that the President and the Committee should be chosen by chance, rather than by ballot. The delegates had been allocated sequential membership numbers of arrival — George, of course, being No. 1. He proposed that one number be chosen at random by the computer — that member would be the President. All members whose numbers divide exactly into the President’s number would be on the Committee. Apathy reigned — this totally undemocratic procedure was agreed.

The computer produced an odd membership number for the President and the number of committee members, including George and the President, was an odd square greater than 10.

What was the President’s membership number?

[enigma1102]

3 July 2017

Posted by on **From New Scientist #2264, 11th November 2000** [link]

George was nominated for president of the Golf Club. There was only one other candidate, and the president was elected by a simple ballot of the 350 members, not all of whom in fact voted.

The ballot papers were taken from the ballot box one at a time and placed in two piles — one for each candidate — with tellers keeping a count on each pile.

George won (what did you expect?), and furthermore his vote was ahead of his opponent’s throughout the counting procedure.

“That must be a one-in-a-million chance,” said the demoralised loser.

“No,” said George. “Now that we know the number of votes we each received, we can deduce that the chance of my leading throughout the count was exactly one in a hundred.”

How many members did not vote?

[enigma1108]

5 June 2017

Posted by on **From New Scientist #2268, 9th December 2000** [link]

George is building a patio, which will be covered using one-foot-square concrete slabs of seven different colours. He has divided the rectangular patio into seven rectangular zones, without any gaps.

Each zone will be covered by slabs of one colour, with five different colours appearing around the perimeter of the patio, and four different colours at the corners. The seven rectangular zones are all different shapes, but all have the same perimeter, which is less than 60 feet.

What are the dimensions of the patio that George is building?

This puzzle is referenced by **Enigma 1221**.

[enigma1112]

24 April 2017

Posted by on **From New Scientist #2274, 20th January 2001** [link]

George is planning to celebrate the new millennium — the

realone — by visiting Foula, the most remote of the Shetland Islands. It is one of the few places in the world where the inhabitants still live by the old Julian calendar rather than the now almost universal Gregorian calendar.In order to correct the drift of the Julian calendar against the seasons, Pope Gregory decreed that in 1582, Thursday 4th October (Julian) should be immediately followed by Friday 15th October (Gregorian), and in order to prevent a recurrence of the drift, years divisible by 100 would henceforth only be leap years if divisible by 400. Previously

allyears divisible by four were leap years. Catholic countries obeyed immediately, others — apart from Foula — fell into line in later centuries.While planning his visit George programmed his computer to print 12-month calendars for the required year, showing weekdays, under both Julian and Gregorian styles. But when he ran the program he was surprised to find that the two printouts were identical.

He then realised that he had entered the wrong year number — the Julian and Gregorian calendars for the year 2001 are

notthe same.What is the first year after 1582 for which they

arethe same?

[enigma1118]

26 March 2017

Posted by on **From New Scientist #2278, 17th February 2001** [link]

While waiting for a much-delayed train, George found himself trying to read a very boring book. He soon gave up and started counting its pages instead. Chapter 1 started on Page 1, and each subsequent chapter started at the top of a page. The boredom factor was enhanced by the fact that the length in pages of each chapter was equal to the chapter number multiplied by the length of Chapter 1.

With still no sign of the train, George proceeded to total all the page numbers in each chapter. Again, the totals for each chapter were exact multiples of the total for Chapter 1, but this time the multiples did not equate to chapter numbers. For the last chapter the multiple was a prime number, even though the chapter number was not.

How many pages were there in the book?

[enigma1122]

3 March 2017

Posted by on **From New Scientist #2282, 17th March 2001** [link]

George has been winning free drinks at his local pub using a trick with four non-standard dice. Each face of each die is marked with one of the numbers 1 to 9, not necessarily all different. One of the nine numbers does not appear on any die, but each die has the same total of its six faces.

George allows you to choose one die, then he chooses one of the others. The two selected die are thrown simultaneously, and the one who throws the smaller number buys the drinks. Draws are impossible.

His friends have discovered that if they choose the red die, George chooses the yellow — if they choose yellow, George chooses green — if they choose green, George chooses blue — and if they choose blue, George chooses red! George expects (statistically) to win exactly two throws in every three with any of these pairs of dice.

We can conveniently represent the markings on a die as a six-digit number, with the digits in ascending order. You can check that 334455 beats 222288 two-to-one, but George’s set does not include either of these dice. The red die includes at least one lucky seven. There is only one set of four dice which will do the trick.

List the six numbers for each of the four colours.

[enigma1126]

16 January 2017

Posted by on **From New Scientist #2288, 28th April 2001** [link]

The PIN code on George’s cash card is a semi-prime number, that is to say it is the product of two different prime numbers. He discovered some time ago that the PIN code multiplied by his car registration number gives his six-digit phone number, which does not begin with a zero. But George has now discovered a slightly more obscure coincidence. If he subtracts his house number from his phone number and multiplies the result by his house number, the result is his phone number with its digits in reverse order!

What is George’s car registration number?

[enigma1132]

26 December 2016

Posted by on **From New Scientist #2291, 19th May 2001** [link]

The Ancient Greeks studied “harmonic triads” of integers, the simplest of which is {2, 3, 6}. The middle number is known as the “harmonic mean” of the other two and is calculated so that its reciprocal is the mean (“average”) of the reciprocals of the other two numbers. George has constructed a “harmonic square” — nine different integers in a 3 × 3 grid such that in each of the three rows, the three columns and the two diagonals, the three numbers form a harmonic triad with the harmonic mean in the middle.

Using his computer, George has discovered that the smallest number which can appear in such a square is 210. You should need only progressive intuition to deduce the other eight numbers which accompany 210 in a harmonic square.

What is the largest of these numbers?

[enigma1135]

21 November 2016

Posted by on **From New Scientist #2296, 23rd June 2001** [link]

George lives in a long road in which the houses are numbered from one with no numbers missing. He has calculated that the total of all the house numbers less than his is equal to the total of all the house numbers greater than his.

George’s brothers, Dave, Ernest and Fred, live in shorter roads than George, but they can each make the same claim regarding house numbers. The brothers’ four house numbers have different numbers of digits.

Hearing this story, George’s drinking friend scribbled on a beer mat for a while, then he asked: “George, does your road have nearly 10,000 houses?”

“No, not nearly that many,” George replied.

How many houses are there in total in the four roads? And what answer would the man in the pub have given to that question before being corrected?

[enigma1140]

10 October 2016

Posted by on **From New Scientist #2302, 4th August 2001** [link]

Reading an old question paper in

Mathematical Challenge, a problem-solving competition in Scotland, George found the question:“Which integer can be multiplied by 99 by appending a single digit 1 at each end?”

While struggling with this question, he discovered that 77 can be multiplied by 23 by adding 1 in front and 1 behind:

77 × 23 = 1771.

With rather more effort he found:

52631579 × 29 = 1526315791.

The answer to the Scottish challenge is: 112359550561797752809.

George eventually proved that 23 and 29 are the smallest prime numbers by which other suitably chosen numbers can be multiplied simply by adding ones fore and aft.

What is the largest prime number by which some other number can be multiplied using this trick?

[enigma1146]

19 September 2016

Posted by on **From New Scientist #2305, 25th August 2001** [link]

The Ancient Egyptians could not comprehend fractions with numerators greater than 1, such as 8/11. If required to divide 8 sacks of wheat evenly among 11 bakers, each could first be given 1/2 a sack, then from the residue each could be given 1/5 of a sack, but not as much as 1/4. With modern arithmetic we can use this logic to calculate that 8/11 = 1/2 + 1/5 + 1/37 + 1/4070.

There are several ways of representing 8/11 as the sum of reciprocals without including such unfriendly fractions as 1/4070. George’s friend Henry has discovered that 8/11 = 1/2 + 1/6 + 1/22 + 1/66, which he likes because the reciprocal denominators are all even divisors of his house number, 66. George has matched this achievement by expressing 8/11 as the sum of the reciprocals of a set of different odd divisors of his odd house number, which is less than 200.

What is George’s house number?

[enigma1149]

15 August 2016

Posted by on **From New Scientist #2310, 29th September 2001** [link]

After the Apathy Party swept into power in the General Election, George, its founder, realised one of the hazards of government: impractical proposals are liable to become law because no one has properly assessed the consequences. The latest is so bizarre that even the Apathy Party must reject it?

The Chancellor of the Exchequer thinks it would be fun to abolish the current coinage and mint just one denomination of coin and one of note. He has proposed an absurd pair of values, each a whole number of pence. George realises that although a large sum of money, such as £999.99, can be paid exactly in several different ways, there are precisely 10,000 amounts that can each be paid exactly using only one combination of the proposed notes and coins. Worse, there is a smaller, but still substantial, number of amounts that cannot be paid exactly using any combination of one or both of the denominations.

How many different amounts cannot be paid exactly?

See also **Enigma 1194**.

Thanks to Hugh Casement for providing a transcript for this puzzle.

[enigma1154]

18 July 2016

Posted by on **From New Scientist #2314, 27th October 2001** [link]

George quickly solved the popular magic square puzzle which asks you to arrange the numbers 1 to 16 in a 4 × 4 grid so that the four rows. the four columns and the two diagonals all have the same sum — so he tried to be different. He has now found an “Anti-Magic Square”, using the numbers 1 to 16, but the ten totals are all

different. They are in fact ten consecutive numbers, but in no particular sequence in relation to the square grid.One of the diagonals in George’s square contains four consecutive numbers and the other contains four prime numbers, each in ascending numerical order from top to bottom.

One row contains four numbers in ascending numerical order from left to right.

What are those four numbers?

**Enigma 8** was also about anti-magic squares.

[enigma1158]

27 June 2016

Posted by on **From New Scientist #2317, 17th November 2001** [link]

In a previous

Enigma, George asked you to find the smallest number which can be multiplied by four simply by moving the last digit to the front. The answer is 102564 × 4 = 410256. In fact, 102564 is the smallest number which can be multiplied by any integer by moving its last digit to the front.George is now trying to find numbers which can be

dividedexactly by some integer (any integer greater than one) simply by moving the last digit (which must not be zero) to the front. He has found two solutions to this puzzle and checked them on his eight-digit calculator. He is now looking for another.What is the smallest number which George has not yet found which can be divided by some factor using this trick?

[enigma1161]

30 May 2016

Posted by on **From New Scientist #2321, 15th December 2001** [link]

The darts players at George’s local have devised a seeding system for themselves. Each has determined the number of darts he needs to throw, on average, in order to hit the Bull once. This is his

— the lower the better. Thanks to remarkable foresight in their parents’ choices of names, the seedings proceeded alphabetically from Alan at No. 1 to Zak at No. 26, with no duplication of initial.ratingFred and George met in the pub recently and decided to throw alternately, one dart at a time, until one hit the Bull. The other would then pay for the drinks. Although Fred is the more accurate darts player, they calculated from the ratings that if George threw first they would each have an even chance of winning the sudden-death game.

Duly refreshed, they then calculated from the ratings that this would be an even-money game for

pair of players who are adjacent in the seedings, provided the lower seed throws first. But if Zak played the game against Alan and threw first, his chance of winning would be exactly 10 per cent.anyWhat is George’s rating?

[enigma1165]

2 May 2016

Posted by on **From New Scientist #2325, 12th January 2002** [link]

George lives not far from the Crucible Theatre, where the annual World Snooker Championship has been held for the past 25 years. Each year 32 players compete in a straightforward knockout tournament to determine who is the champion.

George, and his brother Fred, have entered a less prestigious tournament at their local pub. Since they cannot be certain that the number of entrants will be a power of two, lots will be drawn to see who is awarded byes in the first round. The tournament will then continue as a knockout, also drawn at random.

While waiting for the draw to be made, George calculated the chance that he will at some stage have to play against his brother, assuming all the players are of equal ability. It is exactly one in ten.

How many players entered the tournament?

[enigma1169]

11 April 2016

Posted by on **From New Scientist #2328, 2nd February 2002** [link]

George’s local council celebrated National Plant a Tree Year by planting four saplings in a public park. Unfortunately, a few nights later, vandals dug up one of the trees and cleared the ground leaving no evidence of where the tree had been. George has been asked to help.

The Mayor explained that the straight-line between the six pairs of trees had been carefully measured so that there were only two

differentdistances. With three trees remaining where they had been planted, he asked it George could work out the original position of the fourth.“No,” George replied, “I have identified four possible positions and I am still thinking”.

How many possible positions were there?

[enigma1172]

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