I’m on holiday in the lovely country of Philitaly, and planning to send plenty of postcards because postage is very cheap. But the country only allows up to three stamps on any letter.

Can you tell me which three denominations of stamps would allow me me to cover any cost of postage from 1 cent to 15 cents inclusive?

And which four stamp denominations would allow all values from 1 to 24 cents?

]]>Darts player Juan Andred has noticed that on a standard dartboard, there are some neighbouring pairs of numbers that add up to a square number. For example, 20 and 5 make 25, while 6 and 10 add up to 16. He has been wondering if he can come up with a new arrangement of the numbers 1 to 20 so that all neighbouring pairs add up to a square number. And he has nearly succeeded.

He has 20 at the top of the board, and every pair of neighbours adds to a square — with one exception. On his new board, 18 doesn’t form a square with its clockwise neighbour, which is 15, or with its anticlockwise neighbour.

What does Juan’s “square” dartboard look like?

]]>Harry and Tom have been investigating sets of positive integers that form arithmetic progressions where all the members of the set are prime numbers.

When they looked for a set of five such prime numbers, Harry found the set whose final (largest) prime is the smallest possible for the final prime in such a set. Tom found the set whose final (largest) prime is the next smallest possible.

Exactly the same thing happened when they looked for a set of six such prime numbers, and again when they looked for a set of seven such prime numbers.

What were the smallest and the largest primes in each of the three sets that Tom was able to find?

]]>George and Martha have worked on separate departments of a company which has four-digit telephone extensions. George looked at his extension and it was abcd. Martha’s (larger) also had a, b and c as her first three digits but not necessarily in that order. Her last digit was e. They added up their two four-digit numbers and found that the least significant digit was f. They then looked at the difference and that was a four-digit number of which the least significant digit was g. They then looked at the product and the least significant digit was h. They then looked at the average of the extensions; in it the first two digits were equal, the last two digits were also equal, and the least significant digit was i. I have thus mentioned nine digits, all positive and unequal.

What was Martha’s extension?

]]>Using her special recipe, so that each cubic inch of baked cake weighed one ounce, Mam made the cake for my eighth birthday party. It had a regular octagonal flat top and base, and equal square vertical faces, several inches high exactly.

Not all (but a majority) of the dozen invited pals came to the party. We each had an equal portion of cake (the largest whole number of ounces possible from it, a two-figure number). Mam had the leftover cake. Curiously, if one more or one fewer pal had turned up and our portions had been worked out in the same way, Mam’s leftover would have been the same in each case, but less than she actually got.

How large, in ounces, was my portion?

]]>I have a large box of toy building bricks. The bricks are all cubes (the same size), and can be pushed together then dismantled.

I decided to build the largest cube possible by leaving out all the interior bricks. When my hollow cube was finished I had two bricks left over. I put all the bricks back in the box and gave it to my two children. Each in turn was able to use every brick in the box to construct two hollow cubes, again with all interior bricks removed. Their cubes were all different sizes.

I told them this would not have been possible had the box contained any fewer bricks.

How many bricks were in the box?

]]>Wu, Xi, Yo and Ze had different two-figure numbers of sheep and kept them in a walled field divided by fences into a fold each. Maths whizz, Wu, with the largest flock, noticed that together her flock and Ze’s equalled Xi’s and Yo’s combined; and that, as a fraction, the ratio of Yo’s flock to Xi’s had consecutive upper and lower numbers (eg 3/4), whereas her flock to Xi’s ratio had those numbers swapped over (eg 4/3).

Overnight, storm-damaged fences led to the same number of sheep in each fold. Wu’s old maths’ teacher, told just this number and the above relationships, couldn’t be certain how many sheep Wu owned (which would have been true, also, if he’d been told either fraction instead).

How many sheep did Wu own?

]]>I have written down four different numbers. The third number is the highest common factor of the first two (ie it is the largest number that divides exactly into both of them). The fourth number is the lowest common multiple of the first two (ie it is the smallest number that both of them divide exactly into).

I can consistently replace digits by letters in my numbers so that the highest common factor is HCF and the lowest common multiple is LCM.

What are the first two numbers?

]]>George and Martha have a digital clock, which displays time with six digits on the 24-hour system, ie hh:mm:ss.

One afternoon, George looked at the clock and saw a six-digit display involving six different positive digits. He dozed off immediately, and when he awoke in the evening he saw another display of six digits, again all positive and different. He dozed off immediately and later on (before midnight) he awoke, having slept for exactly 23 minutes longer than the previous time. At that time, he saw a third display, yet again six different positive digits. He thus had seen eighteen digits and the nine positive digits had each appeared exactly twice.

At what time did George wake up after his first sleep?

]]>The sum of the ages of six sisters known to me is 92. Though there is no single whole number greater than 1 that simultaneously divides the ages of any three of them, I did notice this morning, while they lined up for the ski lift, that the ages of any two of them standing one behind the other, had a common divisor greater than 1.

In increasing order, how old are the six sisters?

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