Alan and Cat live in a city which has a regular square grid of narrow roads. Avenues run west/east, with 1st Avenue being the furthest south, while Streets run south/north with 1st Street being the furthest west.

Cat lives at the intersection of 1st Street and 1st Avenue, while Alan lives at an intersection due northeast from Cat. On 1 January 2018, Cat walked to Alan’s house using one of the shortest possible routes (returning home the same way), and has done the same every day since. At first, she walked a different route every day and deliberately never reached an intersection where the Street number is less then the Avenue number. However, one day earlier this year she found that she could not do the same, and repeated a route.

What was the date then?

]]>I recently bought a number of equally priced bags of sweets for a bargain price, spending more than 50p in total. If they had been priced at 9p less per bag, I could have had 2 bags more for the same sum of money. In addition, if each had cost 12p less than I paid, then I could also have had an exact number of bags for the same sum of money.

How much did I spend in total on the sweets?

]]>I have just been looking at the top six best-selling books listed in today’s paper. They are numbers from 1 (for the best selling book) to 6, and after each book its position in last week’s list is given.

It turns out that the same six books were in the list last week. For each of the books I multiplied the number of last week’s, and I got six different answers.

Now, if you asked my the following questions — then you would get the same answer in each case:

1. How many primes are there in that list of six products?

2. How many perfect squares are there in that list of six products?

3. How many perfect cubes are there in that list of six products?

4. How many odd numbers are there in that list of six products?

5. How many of the six books are in a higher place this week than last?

For the books 1-6 this week, what (in that order) were their positions last week?

]]>Four football teams are to play each other once. After some of the matches had been played a document giving a few details of the matches played, won, lost and so on was found. This time I am glad to say that, although it was rather a mess, all the figures given were correct. Here it is:

Team | Played | Won | Lost | Drawn | Goals For | Goals Against | Points |
---|---|---|---|---|---|---|---|

A |
1 | 4 | 5 | ||||

B |
1 | 6 | 1 | ||||

C |
12 | ||||||

D |
9 | 5 | 4 | ||||

E |
1 | 3 |

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

Find the score in all the matches that had been played.

]]>Published June 9 2019 (link)

After a day’s filming, a group of those involved in the film’s production went for a gallop.

They split into threes, with a samurai leading each grouping.

The seven samurai were:

BRONSON, BRYNNER, BUCHHOLZ, COBURN, DEXTER, MCQUEEN and VAUGHN

The others involved in the gallop were:

ALANIZ, ALONZO, AVERY, BISSELL, BRAVO, DE HOYOS, HERN,

LUCERO, NAVARRO, RUSKIN, RUSSELL, SUAREZ, VACIO and WALLACH

For each grouping, any two names from the three had exactly 2 letters in common (eg, BRYNNER and BRAVO have B and R in common).

If I told you who accompanied BRONSON, you should be able to tell me who accompanied (a) MCQUEEN and (b) DEXTER

]]>George and Martha are doing a mathematical crossword. There is a 3 x 3 grid with the numbers 1 to 9 inclusive appearing once each. The clues are as follows:

Across:

top line: a prime number

middle line: a prime number

bottom line: a prime number

Down:

left column: a perfect square

middle column: a perfect square

right column: a prime number

Although you do not need to know this, one of the diagonal three-digit numbers is also prime.

What is the sum of the three “across” numbers?

]]>Beyond The Fields We Know

A field named “Dunsany levels” has four unequal straight sides, two of which are parallel. Anne — with her dog, Newton — walked from one corner of the field straight towards her opposite corner. Leon did the same from an adjacent corner along his diagonal. Yards apart, they each rested, halfway along their paths, where Leon, Anne and a signpost in the field were perfectly aligned. Straight fences from each corner converged at the signpost, making four unequal-area enclosures.

Newton made a beeline for the signpost, on which the whole-number area of the field, in acres, was scratched out. Clockwise, the enclosures were named: “Plunkett’s bawn”, “Three-acre meadow”, “Drax sward” and “Elfland lea”. Anne knew that “Three-acre meadow” was literally true and that “Elfland lea” was smaller by less than an acre.

What was the area of “Dunsany levels” in acres?

]]>Each of the four fields at Sunny Meadows Farm contains some sheep and some cows. On the gate of each field is hung a sign saying what fraction of the animals in that field are sheep. The signs are 1/4, 1/3, 1/2, 2/3.

Farmer Gillian explained that if she exchanged the signs on two of the fields then, by simply moving some sheep from one of the two fields to the other, she could return to a situation where each sign again correctly indicated the fraction of the animals that were sheep in that field.

As I walked round I noticed that the total number of animals on the farm was between 300 and 350.

How many sheep, and how many cows, were on the farm?

]]>Oliver Oddwelly ws a cringing wreck of a man. He frequently needed to have his safe combination number, his bank account number, and his credit card number on hand, but was too frightened to write them down anywhere in case someone found them.

However, one day, just as he was oiling the padlock on the fridge, he suddenly hit upon an ingenious way of concealing the numbers. He decided to compose an arithmetical problem, the solution to which would reveal three seven digit numbers he needed to remember.

The sum he invented is shown below, where the first row added to the second gives the third, the fourth subtracted from from the third gives the fifth, and the fifth added to the sixth gives the seventh. One digit can be erased from each row (not necessarily the same position in each row) and the gaps can be closed up to leave three columns of digits, then a second digit can be rubbed out in the same way to give two columns, then a third to leave one column, so that a valid sum remains each time. The three sets of seven digits erased (read down the columns) respectively reveal the numbers he had to remember. What were the three numbers ?

\[\begin{array}{r} 6897\\ +2968\\ \hline 9865\\ – 4968\\ \hline 4897\\ + 3856 \\ \hline 8753\\ \hline \end{array}\]

]]>“Four-armed is four-warmed,” declared Professor Torqui as he placed the petits fours in the oven in his lab at the Department of Immaterial Science and Unclear Physics.

“There are 4444 of them: a string of 4s. By which I mean, naturally enough, a number in base 10 all of whose digits are 4. Do you like my plus fours? (*) Speaking of 10s and plus fours, you can hardly be unaware of the fact that all positive integral powers of 10 (except 10^1, poor thing) are expressible as sums of strings of 4s.

The most economical way of expressing 10^2 as a sum of strings of 4s (that is, the one using fewest strings and hence fewest 4s) uses seven 4s:

10^2 = 44 + 44 + 4 + 4 + 4

The most economical means of expressing 10^3 as a sum of strings of 4s requires sixteen 4s:

10^3 = 444 + 444 + 44 + 44 + 4 + 4 + 4 + 4 + 4 + 4

“Now, it’s four o’clock, and just time for this puzzle: Give me some-where to put my cake stand and I will make a number of petits fours which is an integral positive power of 10 such that the number of 4s required to write it as a sum of strings of 4s in the most economical way is itself a string of 4s.”

What is the smallest number of petits fours Torqui’s boast would commit him to baking? (Express your answer as a power of 10).

(*) £44-44 from Whatsit Forum.

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