A Grid of Sums: Lower Secondary Mathematics Competition Question

The diagram shows a row of five squares above a row of four circles. The number in each circle is generated by adding the numbers in the two squares immediately above it.

Using only the single-digit numbers 1 to 9, complete the grid so that every number is used once and every circle is the sum of the two squares above it. The diagram has the number 6 already inserted.

This question requires a bit of logic so that you can go through all the possible permutations in a systematic and quick way.

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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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Factorial Primes: Middle Secondary Mathematics Competitions

A factorial prime is a prime number of the form p = n! + 1 or p = n! – 1, where n is a positive integer.

Find the sum of all such distinct factorial primes under 50.

As always, show your solution. Add your answer to the Comments below.

In a competition, without any electronic devices, this question should take about 2 minutes.


Level: Middle Secondary (Yellow)



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Primes in a Sequence: Lower Secondary Mathematics Competition

We have a sequence of numbers defined as follows:

T(1) = 3, T(2) = 4, and T(n) = T(n-1) + T(n-2), where n is a positive whole number greater than 2.

T(1) is obviously a prime number. Find the next five prime numbers that appear in this sequence. Write down their sum.


Answer will eventually be posted below in the Comments section, but best to try it yourself first!


Level: Lower Secondary (Blue)

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Primeless Road: Upper Primary Mathematics Competition Question

Primeless Road is a rather peculiar road. All the odd numbered houses are on one side of the road, and all the even ones are on the other side, both sides starting with the lowest number. Nothing very strange with that! However, none of the house numbers is a prime number.

Assuming each house has exactly one other house directly opposite, what is the number of the house opposite number 20?

You can write your answer and discuss the question in the Comments section below.

Level: Upper Primary (Green)


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Midweek Maths: A Purse Full of Ducats

Three men are sitting round a table. They each have some 1-ducat coins in front of them and there is a purse full of similar coins in the middle of the table.


Alessandro turns to Bruno and says:”If I add the ducats in the purse to my money I will have twice as many ducats as you.” Then Bruno turns to Cesare and says:”If I add the coins in the purse to my money, I will have three times as many ducats as you.” Finally, Cesare says to Alessandro:”If I add the ducats in the purse to those I already have, I will have four times as much money as you.”

How many ducats does each man have and how many are in the purse? Assume the smallest integer solution.

As in the previous question, Fibonacci did not use any algebra to solve this question. For students who feel that algebra is some kind of mental torture, try following a verbal algorithm nearly a page long!

Math-e-Monday: A Merchant from Pisa

This week I am reading the Liber Abaci (The Book of Calculations) by Leonardo Pisano, better known as Fibonacci. The book, published in 1202, contains the famous problem of rabbits breeding in such a way as to generate the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 and so on. However, the Liber Abaci also contains a wealth of other problems, and this week I shall post a small selection of questions inspired by those found in the book.

The question below is fairly easy and can be done by a lower secondary, or even upper primary, student. Interestingly, Fibonacci did not have a well-developed algebra, so his solution is a numerical recipe or algorithm. As part of Fibonacci's promotion of the Hindu-Arabic numerals, he has numerous questions involving fractions. Many such questions are designed to show the ease with which fractions can be manipulated using his 'new' system of numbers compared with using Roman numerals. This is one such question.

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.

Prize Maths Quiz (PMQ3) Announcement for Friday 25 January

The next Gifted Mathematics Prize Maths Quiz shall be PMQ3 and will be posted on Friday 25 January 2013 at 07:03 GMT. The competition will close, as always at 23:59 GMT on the Sunday.

Please use our offical GMT clock in the right-hand column.

Please also read the rules, which have been recently updated and clarified, on our PMQ Prize page.

See You Then!




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