A Squared Prime Problem: Upper Secondary Mathematics Competitions

There are prime numbers p, such that the number squared p² is a factor of [(p-1)! + 1].

For which of the following primes is this true?

2, 3, 5, 7, 11, 13, 17.

In a competition, without electronic devices, this question should take approximately 10 minutes. Doing the question by computer is trivial - although interesting as p becomes larger - the aim here is to find the fastest technique to do it manually, as if it were a test question.

You can post your answer in the Comments below, but please try it first!


Level: Upper Secondary (Red)



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Prize Maths Quiz: A Tricky Tower (PMQ6)

Alice was feeling sick and lying in bed in the middle of the afternoon. She hated feeling sick and she hated lying in bed. Well, that wasn’t entirely true; she loved feeling soft and warm in her bed, but today she was feeling hot and irritated.

Bill suddenly bounced through the door.

“Hello Alice! How are you? I just popped round to cheer you up a bit – Mother said you had a slight fever.”

Bill was being chirpy; it seemed like bad manners to be chirpy on a day when Alice wasn’t.

“Look, I brought you a present!” Bill handed Alice a big box covered in coloured triangles and shapes. Before Alice had a chance to ask, Bill volunteered to explain, “It’s a jigsaw puzzle. A big jigsaw puzzle! I hope you like it.”

Alice thought Bill wanted a pat on the head and a fresh bone. The pattern on the front looked interesting but, as Alice opened the box, she frowned.

Midweek Maths: You are in a maze of twisty little passages, all alike

This week I’ve created my own adventure game. I didn’t mean to, it’s just turned out that way. In the name of finding some amusing (and frustrating) strategic games I headed towards some old favourites, only to discover that they had been dismantled and sold as scrap.

In 1984 Infocom and Douglas Adams released an adventure game version of the Hitchhiker’s Guide to the Galaxy. Employing the same wit and tangential narrative of the original books and radio plays, the game was a huge hit. Then graphics arrived and people forgot how to read and write and the age of text-based interactive fiction, better known as adventure games, became a chapter in geek history.

BBC Radio 4 (yes, radio!) did have a graphical version of this game, which worked about a year ago (or maybe two) but has been terminated. Ironic that pressing the “DON’T PANIC!” button now reveals the laconic phrase “zmachine xml invalid”; this is as terminal as it gets. So, if anybody can find a working version of this game hiding in some corner of the net, then let us know, otherwise it’s back to the Douglas Adams page for the original text version.

Prize Maths Quiz: Trapezium Artist (PMQ4)

Alice was playing out in the garden. She really was getting far too big to be playing on the swing; this made her sad as she liked swinging up as high as she could. However, the creaking noises troubled her and her mind turned to whether she could unhinge the whole structure and fall head-first onto the lawn. It struck her as safer to do this as a theoretical calculation than as an experiment. She knew it could be done, but didn't yet know how.

“Alice! Come on, we’re going shopping!” bellowed her mother. Oh no... why don’t my parents shop on the internet and have it all delivered? Now that’s what supermarkets are for; otherwise they are just dull-markets. And why is my big brother never around when he could be useful? Alice jumped off the swing and rolled around on the grass, making herself look as unpresentable as possible.

“Come on, darling! You know I hate all that traffic later in the day.” Her mother fussed trying to brush off clumps of dried grass from Alice’s dress. That didn’t work!

Shopping malls are ghastly plastic places with very little of interest for Alice. She usually tries to head for the computer shops but they are always full of tall smelly teenagers staring at games. “Alice, I know you love to wander off and I’m not wasting my time looking for you. If we get separated I’ll wait for you in this coffee shop. OK?” Alice was not really old enough to be let out on her own, but she was good at wandering off on her own.

Midweek Maths Puzzle: Heavenly Circles

Detail from Medicina Catholica,
Robert Fludd (1629-31)

Sometimes art or history can provide a spark of inspiration for a puzzle. In this case, we have a detail from a work by Robert Fludd (1574-1637). Fludd was highly influential in Elizabethan England, but picking fights with far better mathematicians such as Kepler and Mersenne did nothing to promote his vision of a kind of mathematical-theology. Indeed, his promotion of a two-worlds theory (terrestrial and celestial, or mundane and heavenly) was precisely the idea attacked by Kepler and Galileo and later refined by Newton.

But enough history, let's look at the picture! Trying to clean up the original I noticed that its negative is far clearer and shows more details, so that's what I've posted. The image shows a large circle at the base, with two medium-sized circles touching each other and the circle beneath them. Finally, the image is crowned by a small circle touching the medium ones.

Math-e-Monday: Half a Triangle

Most of the questions posted so far have been about numbers, so this week we’re going to concentrate on geometry.

The people who set geometry questions in mathematics competition papers like to mess with your eyes! They are not really optical illusions but, rather like a good magician, you can’t see the whole picture because you’re concentrating only on what you’re being shown. Geometric diagrams do usually show you everything you need but not always everything you know. My advice is to redraw the diagram and fill in everything you know that is related to the question; it could be trigonometry or circle theorems or construction techniques. Then make sure you read the question carefully to check for details that are written down but not illustrated in the given diagram.

Lastly, brush up on some of the more obscure geometric theorems. You can always prove things from first principles, but in a maths test that wastes precious time. It is better to be prepared with a basket full of theorems and then figure out which ones you need.

The Question

The diagram above shows a circle centre C (not drawn to scale). The line CQ is perpendicular to the diameter PR. The triangle PQR has double the area of triangle PSR. Find the angle PRS.

This is not a PMQ so feel free to discuss this puzzle in the comments section below.

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Midweek Maths Puzzle: Crossing the Bridge

There are some mathematical questions where the first thing to do is to follow the rules and just ‘play the game’. After a bit of practice you start to figure out the patterns behind the rules. Then you can set about finding a solution.

This is a very old puzzle (old puzzles are old because they remain new, otherwise they’d just be forgotten puzzles) given a fresh lease of life with some computer animation. The rules are simple enough; to get all the people across the bridge in the minimum time. The full rules are on the Crossing the Bridge page.

Math-e-Monday: Prime Combination Lock

Derek has just bought himself a smart new padlock with a combination lock. It is one of those with a 3-digit code that you enter by turning each of the 3 wheels. Derek is playing with it, trying to think of a really clever number that he wouldn’t forget.

He notices that the code was set to 200 and decides to play a game. He picks the middle and right wheels; clicking one of them up a number and the other down a number he gets 291. He continues this until he gets back to 200. Then he tries the same thing with the middle and left wheels, and finally with the left and right wheels.

Derek smiles to himself

a)    Write down the prime numbers that Derek finds using his method.


Derek then wants to find out if he can make every possible 3-digit number. He still starts at 200, turns one wheel up and one wheel down, then writes his new number down. But in his next step he can choose any pair of wheels he wishes, not just the two he started with. At each step he writes down his number so as to keep track.

Prize Maths Quiz: Ice-creams and Cubes (PMQ2)

"Hey, Brenda, finish that PMQ later. Come and look at this!"

Alice and Brenda had spent the day at a flea-market helping their parents sell some old stuff. They were both able to keep the money they made from their own old toys and books. They agreed to meet at a local ice-cream parlour to relax and count their takings. But on arriving, Alice could see her friend had been crying.

"What's wrong Brenda? Why are you crying?"

"Look!" Brenda pointed to her little money box. "Empty! All the money I made today has gone! I don't know what happened, I couldn't just have lost it. My whole day wasted, and can't even buy some new books!"

Alice quickly added up the few coins left in Brenda's money box. She had already counted hers to the nearest penny. Being a bit of a maths whizz, Alice did a quick calculation and said to herself, "Mmm... that's bizarre, my money is exactly the cube of yours; in pennies, of course."

Math-e-Monday: Entangled Triplet

It's Monday. It's been a hectic weekend and our Math-e-Monday questions will be on the lighter side of the spectrum. Here's one you can almost do in your head.

If A x B = 5, B x C = 90 and C x A = 2, given that A, B and C are all positive,
find the value of A + B + C

Do try this yourself before checking the answer.

If you're doing this to help with a maths competition, I'd rate this as Lower Secondary standard; and don't even think about reaching for the calculator app!

Only Three Dice

It's raining outside, and Alice and Bill have to stay indoors. They are looking through Alice's box of toys when she exclaims,"Found it! Let's play Number Quest!" "Sounds like a maths game." says Bill, unenthusiastically. "Yeah, but it's dead easy to play," gushes Alice, "Even for you!" She gave Bill a mocking look that meant he had no choice in the matter. Alice beamed.

They unpacked the game. It was, indeed, a rather simple game; roll the four dice and use any of the four arithmetical operations (and brackets if needed) to make the highest possible number between 1 and 100. The higher the number the more prize money you win and you place your counter on that number so nobody can use it again. But as the game progresses it starts to get harder as many of the top numbers are covered with tokens.

"Oh no!!" shrieked Alice. "I can't find all four dice! I can only find three." Bill thought something really dramatic had just happened, like a spider crawling out of the box. "Maybe we could play something else." he said, trying to put his positive voice to good use. But Alice didn't even reply; she was too busy taking all the games out of her games box to look for the missing die. After a good 10 minutes of rummaging, she admitted defeat.

"Let's play with three dice!!" Alice exclaimed. "Can we still make all 100 numbers, though?" asked Bill. "Probably not," Alice said thoughtfully, "But let's find out!" Bill was just not going to get out of playing Number Quest.

The Question

What are the two smallest numbers that cannot be made with just three 6-sided dice using just arithmetic operations?

Also, what are the two largest numbers that cannot be made?

This is not a Prize question, so feel free to comment.

Good Luck!


Maths Questions posted on Mondays, Wednesdays and Saturdays.
Prize Maths Questions (PMQs) posted on Fridays.

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Don't peek at the answers!

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If you're feeling tired, defocus your eyes on this stereoscopic photograph.

From Wikimedia [public domain]

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