Enigmatic Code

Programming Enigma Puzzles

Category Archives: enigma

Enigma 449: He who laps last

From New Scientist #1600, 18th February 1988 [link]

“This is Hurray Talker reporting from Silverhatch on the 40-lap Petit Prix. The leaders, Hansell and Bisquet, are each going round the circuit in an incredible 59 seconds, except when they have made a pitstop. They were neck and neck for the first seven laps but then things started to go wrong. However, each pit stop has been kept down to an amazing 23 seconds. Here they come now to cross the line and complete another lap. Hansel – now. Biscuit – now. That’s just a 2-second interval. There they go off on the next lap.”

How many laps have Hansell and Bisquet completed?

[enigma449]

Advertisements

Enigma 1061: Par is never prime

From New Scientist #2217, 18th December 1999

The local golf course has 18 holes; each of them has a par of 3, 4 or 5; no two consecutive holes have the same par. If you play the holes in order from 1 to 18 and score par for each hole, the number of strokes that you have played after you have completed any hole is never a prime number.

If I told you the par for one particular hole you could deduce with certainty the par of each of the first 15 holes.

Question 1: What is the number of the hole whose par I would tell you? And what is par for that hole?

If I told you the number of the hole whose par I was going to tell you so that you could deduce with certainty the par for each of the last three holes you could in fact make that deduction even before I had told you the par for that hole.

Question 2: What is the number of the hole whose par I would be going to tell you? And what is par for that hole?

[enigma1061]

Enigma 448: Spoiling the division

From New Scientist #1599, 11th February 1988 [link]

In the following division sum, some of the digits are missing, and some are replaced by letters. The same letter stands for the same digit wherever it appears.

Find the correct sum.

[enigma448]

Enigma 1062: Christmas present

From New Scientist #2218, 25th December 1999

Joseph the carpenter used to cut out rectangular blocks of wood which his young son Jesus would paint. The blocks always had whole number dimensions. They used to say a block was fair if the numerical values of its volume and its surface area were the same, for example the 4×5×20 block was fair as it had volume and surface area both equal to 400. They felt that with a fair block the both did the same amount of work.

As Jesus’s birthday was coming up, Joseph asked him to choose a number and he would try and cut a fair block with volume equal to that number. Jesus chose 2000 which so surprised Joseph that he asked Jesus if he thought people would remember him on his 2000th birthday. Jesus thought for a while then replied that it was hard to say, as it depended on so many things.

Can Joseph cut a fair block with volume 2000? If he can, give its dimensions. If he cannot, give the dimensions of the fair block with volume nearest to 2000.

[enigma1062]

Enigma 447: Secret society

From New Scientist #1598, 4th February 1988 [link]

Six boys from my class have joined together to form a secret society. The each have a different three-digit number, but each of the six numbers uses the same three digits in some different order.

The boys have noticed that, for any two of them, their numbers have a common factor larger than 1 precisely when their names have at least one letter in common. So, for example, Tom’s number and Sam’s number have a common factor larger than 1, whereas Bob’s and Tim’s numbers do not. Ken’s number is prime.

The sixth member of the society is one of Ian, Ben, Rod, Rob, Jak, Vic and Pat.

Who is the sixth member, and what is Bob’s number?

[enigma447]

Enigma 1063: Christmas star

From New Scientist #2218, 25th December 1999

When I was at school I was given a Christmas puzzle to do. So, as far as I can remember it, I’ve reproduced it for you to try:

“Four different numbers larger than 6 have been placed in some of the circles of the Christmas star:

Put the numbers 1 to 6 in the remaining circles (one of them in each) so that the four numbers on each straight line add up to the same total.”

Now that I’ve tried this again I realise that I’ve made a mistake somewhere, because the puzzle as stated is impossible. In fact, it turns out that my only error is that one of the four numbers which I have placed on the star is incorrect.

Which one is incorrect, and what should it be?

Thanks to Hugh Casement for providing the sources for a large number of Enigma puzzles originally published between 1990 and 1999, including this one.

[enigma1063]

Enigma 446: Pocket money

From New Scientist #1597, 28th January 1988 [link]

The benefactor Lord Elpis was superstitious to a degree which surpassed mere triskaidekaphobia, shunning black cats and saluting magpies. Indeed, his superstition was more a form of sympathetic magic. He kept two watches, Tick and Tock. When Tick ran down he would wind Tock, so that Tick could rest and vice versa.

One of his many peculiarities related to money, which he only ever carried in his trouser pockets. His trousers had two pockets and two pockets only, and in these he would carry only those non-zero sums of money which could be split between the two pockets in such a way that the amount in his left pocket multiplied by the amount in his right pocket was exactly equal to the amount in the left and right pockets taken together.

Thus, for example, he could carry £6.25, as it was possible to put £5 of this in his left pocket and the remaining £1.25 in his right, since the product of these sums is equal to the sum of these sums.

Given that 100 pence equals £1 and that the penny is the smallest unit of currency:

(a) How many different sums can Lord Elpis carry?
(b) What is the most he can carry at any one time?

[enigma446]

Enigma 1064: Low score draw

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]

Enigma 445: Court order

From New Scientist #1596, 21st January 1988 [link]

The draw for the Humbledon Ladies Tennis Tournament was as follows:

Enigma 445

After the tournament the umpire noticed that six ladies each lost to the lady one place below them in the above list, three ladies each lost to the lady one place above them, one lady to the lady two places below, one to the lady two places above, one to the lady three places below, one to the lady three places above, one to the lady five places below, and one to the lady six places above.

Who won the tournament and whom did she defeat in the final?

[enigma445]

Enigma 1065: Cute cubes

From New Scientist #2221, 15th January 2000

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

TEN is two away from a perfect cube

and:

there are TEN cubes not more than THOUSAND.

What is the numerical value of THEN?

[enigma1065]

Enigma 444: Rows and rows

From New Scientist #1595, 14th January 1988 [link]

Below is an addition sum with letters substituted for digits. The same letter stands for the same digit wherever it appears, and different letters stand for different digits.

Write the sum out with numbers substituted for letters.

[enigma444]

Enigma 1066: Members of the clubs

From New Scientist #2222, 22nd January 2000

There are only 10 people on Small Island. However, there are many clubs, each consisting of the people with a particular interest. The island’s government will give a grant to any club with more than half the population as members. There are 12 such clubs.

The government wants to set up a committee of two so that every one of the 12 clubs has at least one member on the committee. This afternoon, the government is to look at the 12 membership lists and try to find 2 people to form the committee.

(1) This morning, before it sees the membership lists, can the government be certain that it will be able to find 2 people this afternoon?

There are 1000 people on Larger Island. The situation is similar to Small Island, except that there are 50 clubs that each have more than half the population as members. Also, the government wants to set up a committee of five so that every one of the 50 clubs has at least one member on the committee.

(2) Before the government sees the 50 membership lists, can it be certain that it will be able to find 5 people to form the committee?

The situation on Largest Island is similar to that on Larger Island, except that there are 1 million people.

(3) Before the government of Largest Island sees its 50 membership lists, can it be certain that it will be able to find five people to form its committee?

[enigma1066]

Enigma 443: The bells they are a-changing

From New Scientist #1594, 7th January 1988 [link]

When bells are played in a particular sequence, a “change” is a different sequence obtained from the first by at least one pair of bells which were consecutive the first time reversing their order. Any number of pairs can do this, but no bell is involved in more than one move. So, for example, if four bells are played in the order ABCD, then the possible changes are: ABDC, ACBD, BACD and BADC.

Our local bell-ringing group is very keen. A number of them met last night, including one newcomer. They had a bell each and they rang them in a particular order (with the newcomer ringing first). Then, to test themselves, they decided to write down all the possible changes from that original sequence. They each had a piece of paper and in a few minutes each (including the newcomer) had written down some of the possible changes. They had each written the same number and, between them, they had included all the possible changes exactly once.

They then decided to choose one of these changes to play, but thought they had better choose one in which the newcomer’s bell still played first. So they deleted from their lists all those changes in which the newcomer’s bell had changed places: they all had to delete the same number of possibilities.

Including the newcomer, how many of them were there?

[enigma443]

Enigma 1067: Bye!

From New Scientist #2223, 29th January 2000

A number of players entered for a knockout tennis tournament. Some of them played in the first round games, the rest being given “byes” into the second round so that thereafter there were normal rounds in which all remaining players took part, leading eventually to quarter-finals, semi-finals and the final.

Overall the tournament took a week, with the same number of games being played each day.

Actually a whole-number percentage of the entrants were knocked out in the first round.

What percentage?

[enigma1067]

Enigma 442b: Oh yes I did! Oh no you didn’t!

From New Scientist #1592, 24th December 1987 [link]

After our successful pantomime production in which I played the leading lady, I gave my little costarring helpers some gifts from a big bag of different trinkets, and they each got a different number and none were left.

To make it fairer I gave each helper 10p for each gift that he didn’t get and deducted 40p for each gift that he did get, but that still gave each of them some 10p coins as well as some gifts. It cost me £12.60 in addition to the gifts.

What was the highest number of gifts received by any helper (that little fellow got less than 50p cash)?

What part was I playing?

This puzzle completes the archive of Enigma puzzles from 1987. There is now a complete archive from the start of Enigma in February 1979 to the end of 1987, and also from February 2000 to the final Enigma puzzle in December 2013. Making 1162 Enigma puzzles posted so far, which means there are about 626 left to post.

[enigma442b] [enigma442]

Enigma 1068: Triangular Fibonacci squares

From New Scientist #2224, 5th February 2000 [link]

Harry, Tom and I were trying to find a 3-digit perfect square, a 3-digit triangular number and a 3-digit Fibonacci number that between them used nine different digits. (Triangular numbers are those that fit the formula ½n(n+1); in the Fibonacci sequence the first two terms are 1 and 1, and every succeeding term is the sum of the previous two terms). We each found a valid solution and we each created a second valid solution by retaining two of the numbers of our first solution but changing the other one. Our six solutions were all different.

List in ascending order the numbers in the solution that none of us found.

[enigma1068]

Enigma 442a: Hark the herald angels sing

From New Scientist #1592, 24th December 1987 [link]

“Have a mince pie.”

“Thanks. How did the carol singing go this evening?”

“Very well indeed. There were 353 of us. We started from the village church at 6:00pm and arrived here at the hall some time ago. Have a look at this map here on the wall.

Enigma 442a

“We divided into groups and between us we covered every one of the 12 roads once. For each road, the group would enter at one end, sing the carols as they walked along the road, and leave at the other end. At each of the five junctions, all the groups due to arrive at that junction would come together and then re-divide before setting off again.”

“Did you sing many carols?”

“On each road, every member of the group would sing one verse as a solo. A verse takes one minute to sing.”

“Did people get cold waiting at the junctions for the other groups to join them?”

“No. That is the marvellous thing. We had divided into groups so that, at each junction, all the groups for that junction arrived at precisely the same time. Similarly, all the groups arrived at the hall at the same time.”

“You were very fortunate — a little miracle.”

“Don’t forget,” said the vicar who had been standing with us, “it is Christmas Eve.”

What time did the singers arrive at the hall and what was the total number of verses that they sang?

[enigma442a] [enigma442]

Enigma 1069: Time for elevenses

From New Scientist #2225, 12th February 2000 [link]

I have a cube. On each of its faces is a digit, the style of writing being rather like the display on a calculator. I hold the cube to look at one of its faces from the front and then, keeping the upper and lower faces horizontal, I swivel the cube around and note the digits which I see (all seemingly the right way up) and hence I read off a four-figure number which is divisible by eleven.

Now I repeat the process starting this time looking from the front at one of the faces which was horizontal in the previous manoeuvre. Once again I read off a four-figure number which is divisible by eleven.

Now I start again with the cube in exactly the same position as it was at the start of the first process. This time I keep the left-hand and right-hand faces vertical and I swivel the cube around. Once again I read a four-figure number which is divisible by eleven and also by three odd integers less than eleven.

What was that last four figure number?

[enigma1069]

Enigma 441: The coloured painting

From New Scientist #1591, 17th December 1987 [link]

I looked down at the body slumped over my desk. One hand held my card “Newton Harlowe — Private detective”, and the other a painting. All I knew about painting came from watching my secretary Velda doing her nails. However, I could see in the dim light that is was a 6 × 6 array of small squares, each coloured red or blue or green. As the neon lights on the nightclubs opposite my office window flashed on and off and the light reflected from the wet sidewalks, I was able to make out the vertical columns of the painting. I saw:

though that was not necessarily the order they occurred in the painting. Suddenly the door opened and a raincoated figure with an automatic entered. There was a loud bang and everything went black.

I came round to find myself lying next to the body of a blonde on the floor of a living room. From the sound of the surf outside I could tell it was a beach-house. There on the wall was the painting. The moonlight shone onto it through the shutters. As they moved in the breeze I was able to make out the horizontal rows of the painting. I saw:

though again not necessarily in the right order. Just then a police siren sounded outside. I was going to have to do some explaining, and that painting was the key.

Reproduce the painting.

[enigma441]

Enigma 1070: Time to work

From New Scientist #2226, 19th February 2000 [link]

Amber cycles a distance of 8 miles to work each day, but she never leaves home before 0730h. She has found that if she sets off at x minutes before 0900h then the traffic is such that her average speed for the journey to work is (10 − x/10) miles per hour. On the other hand, if she sets off at x minutes after 0900h then her average speed is (10 + x/10) miles per hour.

(1) Find the time, to the nearest second, when Amber should set off in order to arrive at work at the earliest possible time.

Matthew lives in another town but he also cycles to work, setting off after 0730h, and he has found that his average speed for the journey to work follows exactly the same pattern as Amber’s. He has calculated that if he sets off at 0920h then he arrives at work earlier than if he sets off at any other time.

(2) How far does Matthew cycle to work?

[enigma1070]