Working Systematically - Advanced

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I've submitted a solution - what next?

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

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Introducing NRICH TWILGO

Age

5 to 18

Challenge level

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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Phiddlywinks - a tribute to John Conway

Read this article to find out more about the inspiration for NRICH's game, Phiddlywinks.

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Latin squares

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

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Corresponding Sudokus

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

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LCM Sudoku II

Age

11 to 18

Challenge level

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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Colour islands Sudoku 2

Age

11 to 18

Challenge level

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

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River crossing

Age

11 to 18

Challenge level

There is nothing half so much worth doing as simply messing about in boats...

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Twin corresponding Sudoku

Age

11 to 18

Challenge level

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

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Simultaneous equations Sudoku

Age

11 to 18

Challenge level

Solve the equations to identify the clue numbers in this Sudoku problem.

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All-variables Sudoku

Age

11 to 18

Challenge level

The challenge is to find the values of the variables if you are to solve this Sudoku.

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Crossing the bridge

Age

11 to 18

Challenge level

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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Instant insanity

Age

11 to 18

Challenge level

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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LOGO challenge - pentagram pylons

Age

11 to 18

Challenge level

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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LOGO challenge - sequences and pentagrams

Age

11 to 18

Challenge level

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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LOGO challenge - following on

Age

11 to 18

Challenge level

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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Binomial coefficients

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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Difference dynamics

Age

14 to 18

Challenge level

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

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Sorted

Age

14 to 18

Challenge level

How can you quickly sort a suit of cards in order from Ace to King?

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The Tour de Clochemerle

Age

14 to 18

Challenge level

Can you work out where these 5 riders came in a not-quite-so-famous bike race?

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Plum tree

Age

14 to 18

Challenge level

Label this plum tree graph to make it totally magic!

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Symmetricality

Age

14 to 18

Challenge level

Five equations and five unknowns. Is there an easy way to find the unknown values?

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Parabolic patterns

Age

14 to 18

Challenge level

The illustration shows the graphs of fifteen functions. Two of them have equations $y=x^2$ and $y=-(x-4)^2$. Find the equations of all the other graphs.

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Pole star Sudoku

Age

14 to 18

Challenge level

A Sudoku based on clues that give the differences between adjacent cells.

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Twin chute-swapping Sudoku

Age

14 to 18

Challenge level

A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?

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Bent out of shape

Age

14 to 18

Challenge level

An introduction to bond angle geometry.

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Constellation Sudoku

Age

14 to 18

Challenge level

Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.

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Magic caterpillars

Age

14 to 18

Challenge level

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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A long time at the till

Age

14 to 18

Challenge level

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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Impuzzable

This is about a fiendishly difficult jigsaw and how to solve it using a computer program.

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Function pyramids

Age

16 to 18

Challenge level

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

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Purr-fection

Age

16 to 18

Challenge level

What is the smallest perfect square that ends with the four digits 9009?

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W mates

Age

16 to 18

Challenge level

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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Twin equivalent Sudoku

Age

16 to 18

Challenge level

This Sudoku problem consists of a pair of linked standard Suduko puzzles each with some starting digits

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Ante up

Age

16 to 18

Challenge level

Use cunning to work out a strategy to win this game.

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Exhaustion

Age

16 to 18

Challenge level

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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Adding machine

Age

16 to 18

Challenge level

Can you set the logic gates so that this machine can decide how many bulbs have been switched on?

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Stage 5 cipher challenge

Age

16 to 18

Challenge level

Can you crack these very difficult challenge ciphers? How might you systematise the cracking of unknown ciphers?

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Interpolating polynomials

Age

16 to 18

Challenge level

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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Knights moving

Age

16 to 18

Challenge level

Can you swap the black knights with the white knights in the minimum number of moves?

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Working systematically - advanced