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In this network of 2x3 connected lights, list all achievable states if all lights are initially off. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 12 hours ago • 0 comments

In this network of 3x3 connected lights, initially all off, how many states do we need to check to prove or disprove that we can achieve all possible states of the lights? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 1 day ago • 0 comments

Find, with proof, all k>3 where we can achieve any state of k connected lights in a circle, initially all off. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 2 days ago • 0 comments

Happy Monday! This problem continues from the problem last week. Here are 4 connected lights (flipping a switch changes the state of itself and its neighbours) in a circle, initially all off. Prove that if we can turn on 1 light by itself, we can achieve any state of the lights.

submitted 3 days ago • 0 comments

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submitted 4 days ago • 0 comments

For which natural numbers k is it always possible to turn on all k lights connected in a row regardless of their starting states? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 5 days ago • 0 comments

Starting with k connected lights in a row all turned off, if there are 2 distinct ways (different sets of switch flips) to turn on them all with the fewest number of flips, prove that there is at least starting state of the lights where we cannot turn on all of them. #math #mathsky #ProblemSolving

submitted 6 days ago • 1 comment

Is there a starting state of 5 connected lights in a row where we cannot turn all of them on? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 7 days ago • 0 comments

Prove that i) to turn on all lights with the fewest number of switch flips, each switch can be flipped at most once, and ii) the order of the switch flips makes no difference to the end state of the lights. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 8 days ago • 1 comment

Flipping the switch of one light in this chain of connected lights will change the state (on O or off X) of the corresponding light and its immediate neighbours. Is there a starting state of the 4 connected lights where we cannot turn all of them on? #math #maths #ProblemSolving #mathsky

submitted 9 days ago • 1 comment

Flipping the switch of one light in this chain of connected lights will change the state (on O or off X) of the corresponding light (highlighted) and its immediate neighbours. Continue flipping switches until all lights in the example are on. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 10 days ago • 2 comments

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submitted 11 days ago • 0 comments

If we randomly select 3 vertices of a cube to form a triangle, what are the probabilities that the center of the cube i) is inside, ii) is outside (but on the same plane as), iii) is on the edge of, iv) is not on the same plane as the triangle? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 12 days ago • 0 comments

If we randomly select 4 vertices of a cube to form a tetrahedron, what’s the probability that the tetrahedron is flat (the 4 points are coplanar)? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 13 days ago • 1 comment

If we randomly select 4 vertices of a cube to form a tetrahedron (if the 4 points are coplanar, let’s treat it as a flat tetrahedron.), what are the probabilities that the center of the cube is i) inside, ii) outside, iii) on the surface of the tetrahedron? #math #maths #ProblemSolving #mathsky

submitted 14 days ago • 0 comments

If we randomly select 4 vertices of a regular octahedron to form a tetrahedron (If the 4 points are coplanar, let’s treat it as a flat tetrahedron.), what’s the probability that the center of the octahedron is on the surface of the tetrahedron? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 15 days ago • 0 comments

If we randomly select 3 vertices of a regular polygon with 2n sides to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 16 days ago • 0 comments

Happy Monday! This problem's inspired by the 1973 USAMO Q3. If we randomly select 3 vertices of a regular hexagon to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle? #math #ProblemSolving #mathsky

submitted 17 days ago • 1 comment

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submitted 18 days ago • 0 comments

Is it possible to construct a Lucas Sequence that has numbers with all possible remainders when divided by any finite positive integer? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 19 days ago • 0 comments

Construct a Lucas Sequence that has no number in common with the Lucas Numbers. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 20 days ago • 0 comments

Construct a Lucas Sequence that has no number in common with the Fibonacci Sequence. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 21 days ago • 0 comments

Construct a Lucas Sequence that has no number in common with the sequence from the previous question. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 22 days ago • 1 comment

Construct a Lucas Sequence that contains only numbers that give a remainder of 1 or 3 (infinitely many of each) when divided by 5. #math #maths #ProblemSolving #iteachmath #mathsky

submitted 23 days ago • 1 comment

Happy Monday! This problem is inspired by the 1973 USAMO Q2. A Lucas Sequence is made from two starting terms and a recursive formula for the next term using the two previous ones, like on the left. Construct a Lucas Sequence that contains only odd numbers. #math #ProblemSolving #maths #mathsky

submitted 24 days ago • 0 comments

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submitted 25 days ago • 0 comments

What’s the maximum possible the interior solid angle sum of a tetrahedron? What’s the minimum? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 26 days ago • 0 comments

All triangles have the same interior angle sum of π units (radians). Do all tetrahedrons have the same interior solid angle sum? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 27 days ago • 0 comments

What fraction of a sphere does a vertex of a regular octahedron take up? (Or roughly how many regular octahedrons can we fit around a point?) #math #maths #ProblemSolving #iteachmath #mathsky

submitted 28 days ago • 2 comments

What fraction of a sphere does a vertex of a regular tetrahedron take up? (Or roughly how many regular tetrahedrons can we fit around a point?) #math #maths #ProblemSolving #iteachmath #mathsky

submitted 29 days ago • 0 comments

Using the same unit we developed in the previous question, how big is the interior solid angle at a vertex of a cube? #math #maths #ProblemSolving #iteachmath #mathsky

submitted 30 days ago • 0 comments

Happy Monday! This problem is inspired by Q1 on the 1973 USAMO. If an angle is a portion of a full circle, a "solid angle" is a portion of a full sphere. An angle that covers a full circle is 2π units (rad). How many units should a solid angle be that covers a full sphere? #math #ProblemSolving

submitted 31 days ago • 0 comments

What are the necessary and sufficient conditions on polynomial expressions g(x) and h(x) for their product f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 32 days ago • 0 comments

What are the necessary and sufficient conditions on a, b, c, and d for f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 33 days ago • 0 comments

Is it possible for a=c=1, exactly one of b and d to be an integer, and f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 34 days ago • 1 comment

Is it possible for a=c=1, b, d non-integers, and f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 35 days ago • 3 comments

Is it possible for a, c to be integers, b, d non-integers, and f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 36 days ago • 1 comment

Is it possible for a, b, c, d to all be non-integers, and f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 37 days ago • 3 comments

Happy Monday! This problem is inspired by the 1970 Canadian MO Q10. Consider expanding the expression f(x) = (ax+b)(cx+d). Is it possible for a, b to be integers, c, d non-integers, and f(x) to have all integer coefficients? #math #ProblemSolving #iteachmath #mathsky

submitted 38 days ago • 1 comment

Share your own problem inspired by this week's problem scenario. #math #MathSky

submitted 67 days ago • 0 comments

How many different remainder combinations are there for n integers to have a sum divisible by n? #math #ProblemSolving #MathSky

submitted 68 days ago • 0 comments