I am about to get on the bus, but I don’t have the exact money for the £1 fare. The driver doesn’t give change so I hand over more than £1 and they keep the excess.

Once I have sat down, I realise that even though I didn’t hand over all my coins, the amount of money I had with me was the largest possible amount I could have had in change without being able to pay £1 (or any multiple of £1) exactly.

How much did I have?

[puzzle#52]

]]>

George has 27 small blocks which have been identified with 27 different prime numbers — each block has its number on each face. He has assembled the blocks into a 3×3×3 cube, as shown. On each of the three visible faces, the nine numbers total 320 — but this is not true of the three hidden faces.

George remembers that when he bought the blocks they were assembled into a similar 3×3×3 cube, but on that occasion they showed the same total on each of the six faces, this being the smallest possible total if each block has a different prime number.

What was the total on each face when George bought the blocks?

[enigma967]

]]>On the Mount of Mystery dwell those ostraglobulous giants Og, Gog and Magog. They can smell the blood of any Englishman within 400 leagues. So they spotted John Bull as soon as he came within range. But they let him get as far as the bar at the Pig and Whistle, while they finished their elevenses.

Then away they went. Og strode off in his five league boots, Gog in his six league boots and Magog in his seven league boots. Progress was speedy and John got a shock when he chanced to look though the bottom of his mug ten minutes later. Og was a mere nine leagues away, Gog a mere seven and Magog a mere ten.

Without going into gory details, can you say how far it is from the Mount to the Pig and Whistle?

[tantalizer409]

]]>In the following football table and addition sum digits (from 0 to 9) have been replaced by letters. The same letter stands for the same digit whenever it appears and different letters stand for different digits. The three teams are eventually going to play each other once — or perhaps they have already done so. The teams have been put in the table in alphabetical order.

(Two points are given for a win and 1 point to each side for a draw).

Find the score in each match.

[enigma542]

]]>It is Tom’s 7th birthday and he has a cake with seven candles on it, arranged in a circle – but they are trick candles. If you blow on a lit candle, it will go out, but if you blow on an unlit candle, it will relight itself.

Since Tom is only 7, his aim isn’t brilliant. Any time he blows on a particular candle, the two either side also get blown on as well.

How can Tom blow out all the candles? What is the fewest number of puffs he can do it in?

[puzzle#51]

]]>Albion, Borough, City, Rangers and United have played another tournament in which each team played each of the other teams once. Three points were awarded for a win and one point to each team for a draw.

On points gained, the teams finished in the order: Albion, Borough, City, Rangers, United. Each team gaining a different number of points. Each team won at least one match and lost at least one match.

If I told you the result of one particular match in the tournament you would be able to deduce with certainty the results of all the other matches.

Which particular match, and what was the result?

[enigma968]

]]>In this long division sum, I fear,

Most of the figures simply are not there.

Two and Three and Four and Six,

One of these is wrong. But which?

Three and Six and Four and Two,

Do you think that’s much too few?

Why don’t I give you rather more,

Than Six and Two and Three and Four?

To give you four, you will agree,

Is better than to give you three.

Look at the pattern if you wish.

All the figures look like this:Which figure was wrong? Find the correct division sum.

This completes the *Puzzle* series of puzzles that were originally published in *New Scientist* between May 1977 (when *Tantalizer* finished) and February 1979 (when *Enigma* started).

There is now a complete archive of puzzles from July 1975 up to December 1989, and from March 1998 to December 2013 (when *Enigma* finished). Making a grand total of around 1542 puzzles on the site (plus a few extra) – about 30 years worth!

I will continue posting *Enigma* puzzles on Monday and Friday, and *Tantalizer* puzzles on Wednesday.

[puzzle2]

]]>The following puzzle was sent in by a young schoolboy:

“Five businessmen (Adam, Brian, Clive, David and Edward) all live in Glasgow. On their way to work they all approach the same roundabout from the same direction and each takes a different one of the five possible other exits from the roundabout (labelled A, B, C, D, E in clockwise order from their entry).

The exits lead to Aberdeen, Berwick, Carlisle, Dundee and Edinburgh. In each case the letter of the exit, the initial of the man using that exit, and the initial letter of the destination are all different.

David leaves at an exit labelled with a vowel, Adam leaves the roundabout at the exit before Edward, and the man going to Edinburgh drives past Clive’s exit and leaves three exits later.

What is the label on the Aberdeen exit? Who takes the Dundee exit?”

Unfortunately, this brave attempt does not have a unique answer so I told the young man to resubmit it with an extra clue of the type “Exit ___ does not lead to ___”. He did this and it did then make an acceptable Enigma.

What was the additional clue?

[enigma541]

]]>Petal is on her way to a country fair, to sell some vintage kitchenware belonging to her good friend Gretel. As she walks along the River Biddle admiring the view, Petal trips on a pair of oars left carelessly on the bank. A valuable wrought-iron kettle flies out of her hand and lands on one of the river’s many marshy islands.

From the river bank to the island is 3 metres directly across, and though the oars are sturdy enough to walk on, they are just short of reaching over the water.

How can Petal reach reach the island with the paddles and save Gretel’s metal kettle from its muddy peril?

[puzzle#50]

]]>For the purposes of this

EnigmaI shall define a “palindromic angle” as one whose number of degrees is a whole number less than 180 and is such that the number remains the same if its digits (or digit) are written in reverse order.I have drawn an irregular hexagon and a line across it in a similar fashion to the one shown here, however the given picture is not to scale. In my figure, the eight marked angles are different palindromic angles, and in fact more than two of them are acute.

What (in increasing order) are the six angles of my hexagon?

[enigma969]

]]>It was a fearsome Hogmany, attended by 11 McSporrans, 11 McTavishes, 11 Mackinnons and 11 MacHinerys. Mind you, that makes it sound worse than it was. The total number present was in fact, smaller, each person being a member of exactly two of the clans.

Especially numerous were those who belonged jointly to the McSporrans and McTavishes. There was no other group which outnumbered them and they outnumbered the McSporran-Mackinnons by precisely two to one.

How many of the company were simultaneously Mackinnons and MacHinerys?

The original puzzle spells “Mackinnon” differently each time it is used.

[tantalizer410]

]]>“It takes

all sortsto make a hang-gliders’ convention go with a whirl,” confided Icarus Thorns to the Rev E. B. Inept as they dallied over their tea on a mountain top.“You can say that again,” answered his companion, dipping into the bag of proffered sweets and adding absent-mindedly another two lumps to his already sweetened tea. “You can say that again,” echoed the mountains faintly.

“This puzzle, unlike our jump, needs little preamble. In the (correct) equation:

ABC – D – E – F – G – H – I – J = 100

each of the letters stands for a different digit. All you have to do is deduce the value of:

(D × E × F × G × H × I × J) ÷ ABC .”

So saying, the intrepid Inept threw himself down off the outcrop, his wings glinting in the sun.

Can you deduce the answer before he reaches the bottom?

[enigma540]

]]>I have a snow globe that is a hollow hemisphere. At its highest point, it is 10 centimetres tall, and in the centre stands a tree that is 6 cm tall. How far away from the centre can the tree be moved and still stand upright?

[puzzle#49]

]]>Take a large sheet of lined paper. On line 1 write any number between 0 and 100 (but not necessarily a whole number). In turn, write a number on each of the lines, 2, 3, 4, …, 100, according to the following rule:

Suppose you have just written a number X on a line. If X is less than 50 then write 40 plus half of X on the next line, otherwise write 93 minus half of X.

(A) Can I say now, before you write your number on the 1st line, what the nearest whole number to the number you write on the 100th line will be? If yes, then what is that nearest whole number?

Take a second sheet of lined paper and repeat the above, except that the rule if X is less than 50 is changed; now write 20 plus half of X on the next line.

(B) My question now is as in question A.

Now take a third sheet of lined paper and repeat the procedure, except that the rule becomes the following. If X is less than 50 then write 40 plus half of X, otherwise, write 15 plus half of X.

(C) Once again, my question is as in question A.

[enigma970]

]]>In the addition sum below letters have been substituted for digits. The same letter stands for the same digit wherever is appears, and different letters stand for different digits.

Write the sum out with numbers substituted for letters.

[puzzle3]

]]>In the following division sum each letter stands for a different digit:

Rewrite the sum with the letters replaced by digits.

[enigma539]

]]>The traffic lights near me are annoying: they are green for just 10 seconds and red for 90 seconds. I go through them only on green on my bike every day and I first see the lights as I approach around a bend when I am 15 seconds away. I get upset if I miss a green light that I could have got through. I can speed up by about 25 per cent or I can slow down.

What should my strategy be if the lights are green when I first see them? And what if they are red? And how often might I get upset?

[puzzle#48]

]]>Harry, Tom and I were trying to find a two-digit perfect square, a three-digit perfect square and a four-digit perfect square, such that each of the nine digits from 1 to 9 was used once.

We each found a different valid solution. Both Tom and Harry found the only valid solution for their two-digit squares. Our solutions had no squares in common, but our two-digit squares were consecutive.

List in ascending order the perfect squares used in my solution.

[enigma971]

]]>When H.M.G.’s five top ambassadors all fell under the same bus, there was an immense fuss about their replacements. The corridors of power hummed with insinuations, until at last a list of five names and a set of conditional agreements was achieved. The latter read:

1. If Sir Basil Brace does not get Paris, Sir Emlyn Entry shall have Bonn or Rome.

2. If neither Sir Ambrose Amble nor Sir Donald Duck gets Washington, Sir Crispin Carruthers shall have Paris.

3. If Sir Ambrose Amble does not get Bonn, Sir Crispin Carruthers or Sir Emlyn Entry shall have Paris.

4. If Sir Donald Duck does not get Rome, then, if Sir Ambrose Amble does not get Paris, Sir Emlyn Entry shall have Moscow.

5. If Sir Ambrose Amble does not get Washington, then, if Sir Donald Duck does not get Moscow, Sir Crispin Carruthers shall have Bonn.Having been duly warned that “if x then y” does not mean or imply “if not x then not y”, can you assign the top chaps to the right places?

[tantalizer411]

]]>I recently visited Ruralania with its five towns, Arable, Bridle, Cowslip, Dairy, Ewe, joined by one-way roads labelled high or low, as in the map:

The Ruralanian road system is very simple; if a driver is at one of the five towns then the systems says whether he or she is to leave that town by the high road, by the low road, or to stop in that town.

The system is determined by the fact that it must be possible to make a Grand Tour by starting at a certain town and driving round visiting all the towns and stopping at the final town, while always obeying the single Rule (X):

Take the high road if taking the low road and then obeying Rule (X) again, results in your car next being at A or E. Otherwise, take the low road if taking the high road and then obeying Rule (X) again, results in your car next being at C. Otherwise, stop in the town where you are.

What is the Grand Tour? (List the towns in order).

[enigma538]

]]>