Enigmatic Code

Programming Enigma Puzzles

Tag Archives: by: Richard England

Enigma 1109: All square in games

From New Scientist #2265, 18th November 2000

At tennis a set is won by the first player to win 6 games, except that if the score goes to 5 games all, the set is won either by 7 games to 5 or by 7 games to 6. The first person to win three sets wins the match.

Sometimes at the end of a match each player has won exactly the same number of games. This happened when André beat Boris in a match in which no two sets contained the same number of games.

You will have to work out whether André won by 3 sets to 1 or by 3 sets to 2, but if I told you how many games in total each player won you would be able to deduce with certainty the score in each of the sets that André won and in the set or each of the sets that Boris won.

What was the score in each of the sets that André won? (Give each set’s score in the form x-y, André’s score always comes first).

[enigma1109]

Enigma 1116: United win at last

From New Scientist #2272, 6th January 2001 [link]

Albion, Borough, City, Rangers and United played a tournament in which each team played each of the other teams once. Two matches took place in each of five weeks, each team having one week without a match.

One point was awarded for winning in the first week, 2 points for winning in the second week, 3 points for winning in the third week, 4 points for winning in the fourth week and 5 points for winning in the fifth week. For a drawn match each team gained half the points it would have gained for winning it. At any stage, teams that had gained the same number of points were regarded as tying.

After the first week A led, with B tying for second place. After the second week B led, with C tying for second place. After the third week C led, with R tying for second place. After the fourth week R led, with U tying for second place. After the fifth week U had won the tournament with more points than any of the other teams.

(1) Which team or teams finished in second place after the fifth week?

(2) Give the results of Albion’s matches, listing them in the order in which they were played and naming the opponents in each match.

This completes the archive of Enigma puzzles from 2001. There are now 1065 Enigma puzzles on the site, the archive is complete from the beginning of Enigma in February 1979 to January 1987, and from January 2001 to the final Enigma puzzle in December 2013. Altogether there are currently 59.5% of all Enigmas published available on the site, which leaves 726 Enigmas between 1987 and 2000 left to publish.

[enigma1116]

Enigma 1119: Six primes

From New Scientist #2275, 27th January 2001 [link]

I invite you to select a three-digit prime number such that if you reverse the order of its digits you form a larger three-digit prime number. Furthermore the first two digits and the last two digits of these two three-digit prime numbers must themselves be four two-digit prime numbers, each one different from all the others.

Which three-digit prime number should you select?

[enigma1119]

Enigma 1123: German squares

From New Scientist #2279, 24th February 2001 [link]

In the following statement digits have been consistently replaced by capital letters, different letters being used for different digits:

VIER and NEUN are both perfect squares.

If I told you the number represented by VIER you could deduce with certainty the number represented by NEUN. Alternatively, if I told you the number represented by NEUN you could deduce with certainty the number represented by VIER.

What is the numerical value of the square root of (VIER × NEUN)?

[enigma1123]

Enigma 1129: Minimal change

From New Scientist #2285, 7th April 2001 [link]

The coins from 1p to 100p in circulation are for 1p, 2p, 5p, 10p, 20p, 50p and 100p. If one receives 40p in change for a purchase it can be paid in a minimum of 2 coins (20p + 20p) but could be paid in one coin more than that minimum number (20p + 10p + 10p). But a few amounts of change cannot be paid for in one coin more than the minimum possible number of coins, for instance 1p and 50p.

Harry, Tom and I each paid 100p for an item priced such that the change due could not be paid in one coin more than the minimum possible number of coins. Each item cost a different amount.

Next day Harry and Tom bought the same items at the same price as on the previous day, whereas I bought a different item at a different price from any of the others, which again was such that the change due from 100p could not be paid in one coin more than the minimum possible number of coins. Each day the total cost of the three purchases was a prime number of pence.

How much did each of my purchases cost?

[enigma1129]

Enigma 1136: Triangular numbers

From New Scientist #2292, 26th May 2001 [link]

Triangular numbers are those that fit the formula ½n(n+1), like 1, 3, 6 and 10.

In the following statement digits have been consistently replaced by capital letters, different letters being used for different digits:

“ONE, THREE, SIX and TEN are all triangular numbers, none of which starts with a zero”.

Which numbers are represented (in this order) by ONE, THREE, SIX and TEN?

[enigma1136]

Enigma 1141: Powers of two

From New Scientist #2297, 30th June 2001 [link]

In the following statements digits have been consistently replaced by capital letters, different letters being used for different digits:

TWO is a prime number,
FOUR is a perfect square,
EIGHT is a [perfect] cube.

You should assume that neither TWO nor FOUR nor EIGHT starts with a zero.

Find the number represented by EIGHT.

[enigma1141]

Enigma 1148: Four-way fairway tie

From New Scientist #2304, 18th August 2001 [link]

After they had each played four rounds in the golf tournament Bernhard, Colin, Darren and Ernie all ended up with the same total score even though the scores for the 16 individual rounds that they played were all different. Each player’s score for each round was in the 60s or 70s (that is to say between 60 and 79 inclusive). Bernhard’s score in each of his four rounds was a prime number, Colin’s score in each of his four rounds was a semi-prime (the product of two prime numbers), Darren’s and Ernie’s scores in each of their four rounds were numbers that are neither prime nor semi-prime. Darren’s best (lowest) round was better than Ernie’s best round, and Darren’s worst round was better than Ernie’s worst round.

List in ascending order Ernie’s scores for the four rounds.

[enigma1148]

Enigma 1153: Luconacci numbers

From New Scientist #2309, 22nd September 2001 [link]

In the Fibonacci sequence the first two terms are 1 and 1, and each subsequent term is the sum of the previous two terms; so the sequence starts 1, 1, 2, 3, 5. Less well known is the Lucas sequence, whose first two terms are 1 and 3, and each subsequent term is the sum of the previous two terms; so the sequence starts 1, 3, 4, 7, 11. In the Tribonacci sequence (so named by Mark Feinberg) the first three terms are 1, 1 and 2, as in the Fibonacci sequence, and each subsequent term is the sum of the previous three terms; so it starts 1, 1, 2, 4, 7.

Harry, Tom and I were looking to find a 2-digit Fibonacci number, a 2-digit Lucas number and a 2-digit Tribonacci number that used six different digits. We each found a different solution; our three Fibonacci numbers were all different from each other; our three Lucas numbers were all different from each other; and our three Tribonacci numbers were all different from each other. None of the numbers in my solution appeared in either Harry’s or Tom’s solution.

List in ascending order the numbers in my solution.

[enigma1153]

Enigma 1157: Never the same result

From New Scientist #2313, 20th October 2001 [link]

Albion, Borough, City, Rangers and United have played another tournament. This time each team played each of the other teams twice. The two matches that any two teams played against each other had different results, even in the case of City, which did not draw any of its matches.

Two points were awarded for a win and one point for a draw. After each team had played each of the other teams once, United was in the lead, one point ahead of Rangers, which was one point ahead of City, which was one point ahead of Borough, which was one point ahead of Albion.

But by the end of the tournament the positions were completely the opposite: Albion finished one point ahead of Borough, which was one point ahead of City, which was one point ahead of Rangers, which was one point ahead of United.

If you knew the results of the first match between Borough and United you could deduce with certainty the results of all the other matches played.

Give the result of the first match Borough played against each of the other four teams, naming the opponents in each match.

[enigma1157]

Enigma 1162: Triangular or square

From New Scientist #2318, 24th November 2001 [link]

Harry and Tom each chose a four-digit number that was either a perfect square or a triangular number and told me its last two digits. I deduced that Harry’s number was one of exactly two four-digit perfect squares or one of exactly two four-digit triangular numbers, but that Tom’s number was one of exactly three four-digit perfect squares or one of exactly three four-digit triangular numbers.

Then they told me that the sum of the digits of Harry’s number was the same as the sum of the digits of Tom’s number.

What were (a) Harry’s number and (b) Tom’s number?

[enigma1162]

Enigma 344: Five-nations championship

From New Scientist #1493, 30th January 1986 [link]

England, Scotland, Wales, Ireland and France took part in the Rugby union five-nations championship, each country playing each other country once. Two match points were awarded for a win and one match point for a draw, and the total number of match points gained by each country was: England 7, Scotland 6, Wales 4, Ireland 2, France 1.

No two matches had the same score and no country scored more than six points in any match; but each country scored exactly the same number of points in the championship as each other country, and Wales also had this number of points scored against them.

Those not familiar with the point scoring system of rugby union merely need to know that it is not possible for a side to score one, two or five points in a match.

What were the results and scores of Wales’s matches in the championship?

[enigma344]

Enigma 1168: Thrice unfactorised

From New Scientist #2324, 5th January 2002 [link]

I have found a four-digit number such that it is impossible to factorise the numbers formed by its first digit or last digit or first two digits or middle two digits or last two digits or first three digits or last three digits or all four digits. In other words all those eight numbers are prime except that either or both of the single-digit numbers may be unity.

Harry and Tom have also each found such a four-digit number. The four-digit numbers that we have found are all different; but Harry’s number uses the same digits as Tom’s number, though in a different order.

Which four digit number have I found?

This completes the archive of puzzles from 2002.

[enigma1168]

Enigma 1171: Irregular hexagon

From New Scientist #2327, 26th January 2002 [link]

I drew an irregular hexagon. I also drew three straight lines, each of which ran from one angle of the hexagon to the next angle but two. Since the lines all intersected at the same point the hexagon was divided into six triangles. At the point of intersection each triangle had an angle of 60°.

All sides of all the triangles were an integral number (less than 50) of centimetres in length, and the six sides that met at the point of intersection (each one common to two triangles) all had different lengths.

What in centimetres was the perimeter of the hexagon?

[enigma1171]

Enigma 1175: French squares

From New Scientist #2331, 23rd February 2002 [link]

In the following statement digits have been consistently replaced by capital letters, different letters being used for different digits:

UNNEUF and CENT are all perfect squares.

Find the numerical value of the square root of (UN × NEUF × CENT).

For some reason this puzzle is labelled as Enigma 1112 in the magazine.

This puzzle is somewhat similar to Enigma 1291 (also set by Richard England).

[enigma1175]

Enigma 1178: Last two digits

From New Scientist #2334, 16th March 2002 [link]

The teacher challenged the children to find a particular 5-digit number consisting of 5 different digits such that when they multiplied it by 2 they created another 5-digit number consisting of the other 5 digits.

By giving the children the last two digits of the number that they had to find the teacher ensured that there was only one possible answer; but I shall be less kind and merely tell you that the last digit of the number they had to find was 1.

What was that number ending in 1 that they had to find?

[enigma1178]

Enigma 1182: Recurring decimals

From New Scientist #2338, 13th April 2002 [link]

Some fractions expressed as decimals consist of a decimal point followed immediately by a set of digits that recurs ad infinitum: for example 2/37 = 0.054054054…

In this example the digits that form the fraction and the recurring decimal are all different from each other, but there are only six of them (2, 3, 7, 0, 5, 4).

Your task is to find another fraction written in its simplest form that when expressed as a decimal consists of a decimal point followed immediately by a set of digits that recurs ad infinitum. The digits that form the fraction and the recurring decimal must all be different from each other and there must be nine of them.

What is the fraction?

[enigma1182]

Enigma 1186: Always or never a semi-prime

From New Scientist #2342, 11th May 2002

At snooker a player scores 1 point for potting one of the 15 red balls, but 2, 3, 4, 5, 6 or 7 points for potting one of the 6 “colours”.

Whyte potted his first red, then his first colour, then his second red, then his second colour, and so on until he had potted all 15 reds, each followed by a colour. Since the colours are at this stage always put back on the table after being potted, the same colour can be potted repeatedly.

After Whyte had potted each of the 15 colours his cumulative score called by the referee was always a semi-prime. A semi-prime is the product of two prime numbers; the square of a prime number counts as a semi-prime.

After potting the 15 reds and 15 colours a player tries to pot (in this order) the balls scoring 2, 3, 4, 5, 6 and 7 points. Whyte did so to complete a total “clearance”, but his cumulative score after each of those six pots was never a semi-prime.

What was his final score?

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

[enigma1186]

Enigma 1194: Only two coins

From New Scientist #2350, 6th July 2002 [link]

Harry, Tom, and I have joined the Apathy Party, which as reported in Colin Singleton‘s puzzle Enigma 1154 proposes to reduce the currency to just two denominations. We each want a currency composed of two coins, each a whole number of pence. We have each suggested which two denominations should be chosen.

We have calculated for each of the three suggestions the largest integral amount that cannot be paid exactly with any combination of the two coins, and also the largest amount that can be paid exactly by only one combination of the two coins.

In each case the answer is a two-digit number. Indeed, the largest integral amount that cannot be paid exactly by any combination of my two denominations is the same as the largest amount that can be paid exactly by only one combination of Harry’s two denominations and also the same as the largest amount that can be paid exactly by only one combination of Tom’s two denominations. (A “combination” need not include both denominations).

Harry’s denominations are different from Tom’s.

Which two denominations have I suggested?

[enigma1194]

Enigma 1197: Nombres triangulaires

From New Scientist #2353, 27th July 2002 [link]

Triangular numbers are those that fit the formula ½n(n+1), such as 1, 3, 6 and 10.

In the following statement digits have been consistently replaced by capital letters, different letters being used for different digits:

UN, TROIS, SIX, and DIX are all triangular numbers, none of which starts with a zero.

Which numbers are represented by (in this order) UN, TROIS, SIX and DIX?

See also Enigma 1300.

[enigma1197]