Farmer Quentin keeps his prize bull in a circular field of diameter 50m. To improve the layout, he decides to switch to a square field with side length equal to the old diameter. He builds the new fencing around the old, with fence posts every metre starting at the corner. After completing the new fencing, he tethers the bull outside the fence to one of these fence posts with a rope of integer metre length, to allow the removal of the inner circular fence. As a result, the bull’s temporary roaming area is now four times what it was inside the circle.
What is the length of the rope?
On my travels, I came across a group of some knights (always truthful) and some knaves (always lying). At first, I couldn’t tell which were knights and which were knaves. However, six individuals then gave the following statements about their group: “The number of knights is prime.” “The number of knights is odd.” “The number of knaves is square.” “The number of knaves is even.” “The total of knights and knaves is an odd number.” “There are more knaves than knights.”
These statements enabled me to deduce the composition of the group.
How many (a) knights and (b) knaves were in the group?
My grandson likes making puzzles; his most recent one started with a four-digit number with all the digits different. He added up all the possible two-digit combinations of these four digits (eg, if the digits were 0, 1, 2 and 3 he would add 01+02+03+10+12+…). This total divided exactly into his four-digit number.
Not satisfied with this, he said he had added an extra digit (different from all the others) to the end of his number to make a five-digit number. The five-digit number was an exact multiple of the total of all the two-digit combinations of these five digits. He said that if he told me how many digits were in this multiple, I would be able to work out his five-digit number.
What was his five-digit number?
I recently attended a celebration with fewer than 100 family and friends. We sat round a large oval table with one seat for each person. I said to Liam that he could pick any seat, but it might be interesting to work out how many different arrangements were possible for people round the table. He replied “I’ll leave that to you Grandad”, then (mischievously) added “but remember I always sit next to you and to Granny”. The restriction that I must sit next to Liam divided the number of possible arrangements by a certain prime number. The further restriction that he must also be next to Granny caused further division by a prime number with a different number of digits.
What, in ascending order, are the number of people round the table and the two prime numbers?
I offer my friends a bet: they pay £1 to play, then draw a card at random from a standard 52-card deck. If they draw the Queen of Hearts, I pay them a £30 prize, otherwise I keep their £1.
Although I have many friends, I have a reasonable chance of avoiding paying a prize. But I make a mistake! As I go around the group offering the bet to each friend, I forget to take their chosen card back and shuffle it into the deck, meaning each new bet is made with fewer cards. As a result, my chances of having to pay out are nearly 50 per cent greater. In fact if I had just one more friend, then my chances of having to pay out would be more than 50 per cent greater than if I hadn’t made the mistake.
How many friends did I have in the group?
The teams in our local football league play each other once during the season, getting three points for a win and one for a draw. On the last day of the season each team played a game and then I worked out the league table with the teams in alphabetical order. Here is part of three rows of that league table in which digits have consistently been replaced by letters, with different letters used for different digits.
How many teams are there in the league and what number is LOST?
Secret agent Robert Holmes was searching the hotel room of a foreign agent who was downstairs having breakfast. Holmes discovered a piece of paper containing the text DKCCVTCSZQRZYTAZXTTX and he thought this might be a coded message about the foreign agent’s mission so he sent it to his code-breaking experts.
They discovered that it was a message that had been scrambled by consistently replacing each letter of the alphabet with a different letter (no letter being used to replace more than one different letter). They decoded the message as a sentence containing four words, which they sent back to Holmes with spaces inserted between words. Holmes realised that his life was in imminent danger as soon as he read it.
What was the decoded message?
6oker is a 6-card version of poker. Like 5-card poker the rank of a hand is based on the number of distinct variants of that type possible from a 52-card pack. Fewer variants gives a higher (winning) rank. For example, a running flush (A2345 to 10JQKA in one suit) has 40 variants, beating four-of-a-kind (eg, four aces with another card) which has 624 variants.
Playing 6oker, Al and Di held hands of different rank. Each comprised only two card values and no aces, jacks, queens or kings (eg, four 3s and two 6s). These four values had no common prime factors. Ignoring suits, if you were told just Al’s hand you couldn’t be sure of Di’s, but if you were told just Di’s you could be sure of Al’s.
Who won? Ignoring suits, give Al’s hand.
Having racked my brains all day trying to devise a suitable Teaser, I woke with my mind racing, just after 4am according to my digital clock. I couldn’t get back to sleep, so I decided to try counting sheep. I imagined them to be numbered starting with the figures shown on the clock, then counting upwards. Each sheep jumped through a particular gap in a fence according to the number of prime factors of its number. Repetitions were counted, so that sheep 405 (=3x3x3x3x5) jumped through the fifth gap.
The last thing I remember noticing before eventually falling asleep was that, for the first time, five consecutive sheep had jumped through the same gap.
What was the number of the last of these five sheep?
I have a map showing the location of four castles. All of the distances between the castles are different two-figure whole numbers of miles, as the crow flies. Alton is due north of Sawley; Derry is furthest west. Fenwick is due east of Derry. Alton and Derry are the shortest distance apart, while the distance from Alton to Sawley is the largest possible to comply with all the other information. Again, as the crow flies,
How far is a round trip of the castles (A to F to S to D to A)?