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.

]]>The “value” of a number is found by subtracting its first digit from the last. For example, 6, 72, 88 and 164 have values 0, – 5, 0 and 3 respectively.

Raising funds for a local charity, I placed some raffle tickets numbered from 1 up to a certain 3-digit number, in a box. Participants then selected a ticket at random. If the value of their number was positive, they won that amount in £; if the value was negative, they contributed that amount in £. Otherwise no money changed hands.

All the tickets having been used, the total amount raised in £ was a rearrangement of the digits in that number of tickets.

How much was raised?

]]>Nugel, Chancellor of the Exchequer of Planet-X, was to bring before the Prime Minister the results of his latest calculations of the planetary trade deficit. The calculation had proved awkward (but not as awkward, he thought ruefully, as trying to account for them was going to be) and he had asked the Minister for More Mathematics (who had attended the same school many moons ago) to process the figures in secret on his Megathon computer. The result arrived while the Chancellor was already waiting to be admitted to The Presence an he read the missive at once:

“A boring task, my dear Nugel, but an aesthetic result: your trade deﬁcit in Zloots is represented by the smallest integer which is simultaneously half a perfect square, a third of a perfect cube, a fifth of a perfect fifth power and a seventh of a perfect seventh power.”

Seventh heaven was where Nugel wasn’t when close inspection of the letter failed to reveal the number. He felt let down by his ex-prefect. What on Planet-X was the old coot on about. With a sigh and resourceful to the last, he took out his battered Clackulator and set to work. In the background he could hear the voice of the only minister (Prime or Composite) to have survived 19 terms of office.

Can you find the trade deficit in Zloots (give your answer as a product of powers of prime numbers) before Nugel becomes the ex-Chancellor of the Exchequer?

]]>Adam and Eve have convex hexagonal gardens whose twelve sides are all the same whole number length in yards. Both gardens have at least two right-angled corners and the maximum possible area this allows. Each garden has a path from corner to corner down an axis of symmetry. Adam multiplies the sum of the path lengths by the difference of the path lengths (both in yards) and Eve squares Adam’s answer, getting a perfect fifth power with no repeated digit.

What was Eve’s answer?

]]>I have two accounts at Midloids bank, both with unusual eight-digit account numbers, which are made up of a combination of all odd digits. If either of the account numbers is split in half it gives two four-digit prime numbers. These two primes contain the same four digits, but in a different order, and with no digit repeated. Furthermore if these four-digit primes are split in half, they each give two two-digit prime numbers. If, for both numbers, the prime formed from the first four digits is larger than the prime formed from the second four digits, what are the numbers of my accounts?

]]>Revisiting the old curiosity shop, I bought an unusual moving-iron ammeter (made by Rho Iota Tau Associates). On the non-linear scale “9” was the last possible “whole amperes” scale graduation marked before the “full scale deflection” end stop (which was a half turn of the pointer from zero).

The booklet gave the pointer swing from zero (in degrees) equal to the current (in amperes) raised to a fixed single-figure positive whole number power, then divided by a positive whole number constant. Curiously, the angle between “exact” pointer swings, calculated using this formula, for two single-figure “whole amperes” values was exactly a right angle.

What, in amperes, were these two currents (lower first) and to what power were they raised?

]]>The children at the village school have a number game they play. A child begins by writing a list of numbers across the page, with Just one condition, that no number in the list may be bigger than the number of numbers in the list. The rest of the game involves writing a second list of numbers underneath the first; this is done in the following way. Look at the ﬁrst number, that is, the left-hand end one, as we always count from the left. Say it is 6, then ﬁnd the sixth number in the list – counting from the left – and write that number in the first place in the second row – so it will go below the 6. Repeat for the second number in the list and so on. In the following example, the top row was written down, and then playing the game gave the bottom row:

**6, 2, 2, 7, 1, 4, 10, 8, 4, 2, 1**

**4, 2, 2, 10, 6, 7, 2, 8, 7, 2. 6**

The girls in the school use the game to decide which boys are their sweethearts. For example, Ann chose the list of numbers:

**2, 3, 1, 5, 6, 4**

For a boy to become Ann’s sweetheart he has to write down a list of numbers, play the game, and end with Ann’s list on the bottom row.

Bea chose the list.

**2, 3, 2, 1, 2**

and Cath the list,

**3, 4, 5, 6, 7, 1, 2, 5, 7, 3, 6, 9**

Find all the lists, if any, which enable a boy to become the sweetheart of Ann, of Bea, and of Cath.

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

Rewrite the division sum with the letters replaced by digits.

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