Resources tagged with Making and proving conjectures similar to Euclid's Algorithm I:
Other tags that relate to Euclid's Algorithm I
Making and proving conjectures. Games. Learning through exploration. Interactivities. Visualising. Integers. Complex numbers. Diophantine equations. Euclid's algorithm. Mathematical reasoning & proof.There are 45 results
Polycircles
Stage: 4 Challenge Level: 
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
Tri-split
Stage: 4 Challenge Level:
A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?
Cyclic Triangles
Stage: 5 Challenge Level:
Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
More Beads
Stage: 5 Challenge Level:
With red and blue beads on a circular wire; 'put a red bead between any two of the same colour and a blue between different colours then remove the original beads'. Keep repeating this. What happens?
Epidemic Modelling
Stage: 5 Challenge Level:
Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.
Thebault's Theorem
Stage: 5 Challenge Level:
Take any parallelogram and draw squares on the sides of the parallelogram. What can you prove about the quadrilateral formed by joining the centres of these squares?
Why Stop at Three by One
Stage: 5
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
Triangles Within Squares
Stage: 4 Challenge Level:
Can you find a rule which relates triangular numbers to square numbers?
Triangles Within Triangles
Stage: 4 Challenge Level:
Can you find a rule which connects consecutive triangular numbers?
Triangles Within Pentagons
Stage: 4 Challenge Level:
Show that all pentagonal numbers are one third of a triangular number.
Recent Developments on S.P. Numbers
Stage: 5
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
Vecten
Stage: 5 Challenge Level:
Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.
The Kth Sum of N Numbers
Stage: 5
Yatir from Israel describes his method for summing a series of triangle numbers.
What's Possible?
Stage: 4 Challenge Level:
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Center Path
Stage: 3 and 4 Challenge Level:
Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of. . . .
2^n -n Numbers
Stage: 5
Yatir from Israel wrote this article on numbers that can be written as 2^n-n where n is a positive integer.
Pythagorean Fibs
Stage: 5 Challenge Level:
What have Fibonacci numbers got to do with Pythagorean triples?
Fibonacci Fashion
Stage: 5 Challenge Level:
What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
Loopy
Stage: 4 Challenge Level:
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
Discrete Trends
Stage: 5 Challenge Level:
Find the maximum value of n to the power 1/n and prove that it is a maximum.
Conjugate Tracker
Stage: 5 Challenge Level:
Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.
Least of All
Stage: 5 Challenge Level:
A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.
Trig Rules OK
Stage: 5 Challenge Level:
Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...
Fibonacci Factors
Stage: 5 Challenge Level:
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Golden Powers
Stage: 5 Challenge Level:
You add 1 to the golden ratio to gets its square. How do you find higher powers?
To Prove or Not to Prove
Stage: 5
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Problem Solving, Using and Applying and Functional Mathematics
Stage: 1, 2, 3, 4 and 5 Challenge Level:
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
Cyclic Quads
Stage: 4 Challenge Level:
Points D, E and F are on the the sides of triangle ABC. Circumcircles are drawn to the triangles ADE, BEF and CFD respectively. What do you notice about these three circumcircles?
Fixing It
Stage: 5 Challenge Level:
A and B are fixed points on a circle and PQ is a variable diameter. Make and prove a conjecture about the locus of the intersection of AQ and BP.
Summats Clear
Stage: 5 Challenge Level:
Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.
Pericut
Stage: 4 and 5 Challenge Level:
Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?
Binary Squares
Stage: 5 Challenge Level:
If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
Quad in Quad
Stage: 4 Challenge Level:
The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?
Rotating Triangle
Stage: 3 and 4 Challenge Level:
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
OK! Now Prove It
Stage: 5 Challenge Level:
Make a conjecture about the sum of the squares of the odd positive integers and prove your conjecture.
Polite Numbers
Stage: 5 Challenge Level:
A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
Sixty-seven Squared
Stage: 5 Challenge Level:
Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?
Poly Fibs
Stage: 5 Challenge Level:
A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.
Arrowhead
Stage: 4 Challenge Level:
The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
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}.
Plus or Minus
Stage: 5 Challenge Level:
Make and prove a conjecture about the value of the product of the Fibonacci numbers F{n+1}F{n-1}.
Janine's Conjecture
Stage: 4 Challenge Level:
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
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