Resources tagged with Mathematical reasoning & proof similar to Geometry and Gravity 1:
Other tags that relate to Geometry and Gravity 1
Topology. Investigations. Spheres. Mathematical reasoning & proof. 2D representations of 3D shapes. Logo. Angles. Non Euclidean Geometry. Curvature. Geodesics.There are 167 results
Geometry and Gravity 2
Stage: 3, 4 and 5
This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.
Tourism
Stage: 3 Challenge Level: 
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Marbles
Stage: 3 Challenge Level:
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
More Marbles
Stage: 3 Challenge Level:
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
Three Balls
Stage: 4 Challenge Level:
A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?
Symmetric Tangles
Stage: 4
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Mouhefanggai
Stage: 4
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Euler's Formula and Topology
Stage: 4 and 5
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Konigsberg
Stage: 3 Challenge Level:
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Konigsberg Plus
Stage: 3 Challenge Level:
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Air Nets
Stage: 2, 3, 4 and 5 Challenge Level:
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Yih or Luk Tsut K'i or Three Men's Morris
Stage: 3, 4 and 5 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Long Short
Stage: 4 Challenge Level:
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
Picturing Pythagorean Triples
Stage: 4 and 5
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Whole Number Dynamics V
Stage: 4 and 5
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Happy Numbers
Stage: 3 Challenge Level:
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
A Knight's Journey
Stage: 4 and 5
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Whole Number Dynamics IV
Stage: 4 and 5
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
Magic Squares II
Stage: 4 and 5
An article which gives an account of some properties of magic squares.
Impossible Sandwiches
Stage: 3, 4 and 5
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
DOTS Division
Stage: 4 Challenge Level:
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
Our Ages
Stage: 4 Challenge Level:
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .
Mediant
Stage: 4 Challenge Level:
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Mod 3
Stage: 4 Challenge Level:
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Common Divisor
Stage: 4 Challenge Level:
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Classifying Solids Using Angle Deficiency
Stage: 3 and 4
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
The Frieze Tree
Stage: 4
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Whole Number Dynamics I
Stage: 3, 4 and 5
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
A Biggy
Stage: 4 Challenge Level:
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Ratty
Stage: 3 Challenge Level:
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
Postage
Stage: 4 Challenge Level:
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
Logic, Truth Tables and Switching Circuits Challenge
Stage: 2, 3, 4 and 5
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to. . . .
Reverse to Order
Stage: 3 Challenge Level:
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Prime AP
Stage: 4 Challenge Level:
Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?
Always Perfect
Stage: 4 Challenge Level:
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Power Quady
Stage: 4 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Angle Trisection
Stage: 4 Challenge Level:
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Pythagorean Triples I
Stage: 4 and 5
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Pythagorean Triples II
Stage: 4 and 5
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Sixational
Stage: 4 Challenge Level:
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
Why 24?
Stage: 4 Challenge Level:
Take any prime number greater than 3, square it, subtract one and divide by 24. Make a statement about what you notice about dividing by 24 (a conjecture) and prove that what you say is always true.
Take Three from Five
Stage: 3 Challenge Level:
Can you guarantee that from any group of 5 numbers it is always possible to choose three numbers that will add up to a multiple of 3?
Fitting In
Stage: 4 Challenge Level:
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
Number Rules - OK
Stage: 4 Challenge Level:
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
Whole Number Dynamics II
Stage: 3, 4 and 5
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
Children at Large
Stage: 3 Challenge Level:
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Salinon
Stage: 4 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Seven Squares
Stage: 3 Challenge Level:
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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