Khan Academy Puzzles

If you look at the Khan Academy's math section, you'll see "Math for fun and glory". In the puzzles area, you'll see 3 transformation puzzles and 6 light puzzles. Unfortunately, the hints are relatively useless there and there is no way to get the answer. If you happen to get stuck and want the answer, look below.

## Light Puzzles

For the first three light puzzles, it is possible to either brute force your way through or find the solutions online by searching up "Light Puzzle Solver".

For Light Puzzle x1 and x2, I've discovered that repeatedly pressing squares that have no light will eventually lead you to the solution.

For Light Puzzle x3, it is important to first press the corners (if the grid is relatively empty) and then repeatedly click squares that have no light. Once you've gotten into a "checkerboard" pattern you should be close to completing the puzzle. Sometimes it may take 50+ moves to reach the solution.

## Transformation Puzzle 1

If you need to use dilutions to rotate the fish by 180 degrees, dilate at (0,0) by -1.

If you need to use reflections to rotate the fish by 180 degrees, reflect around the Y and then the X axis.

If you need to dilate a hexagon to translate it 9 units, dilate at (0,0) by 1/2, and then dilate it by 2 at (-9,0).

If you need to rotate a trapezoid, rotate by 180 degrees at (0,0), and then rotate it by another 180 degrees at (6,0).

If you need to use reflections to translate the fish, reflect over the line x=0, and then reflect over x=5.

## Transformation Puzzle 2

For reflecting a quadrilateral about the Y-axis, complete the following steps:

* Rotation by -90° about (0,-2)
* Dilation of scale 1/2 about (0,2)
* Reflection over the line from (-2,0) to (2,0)
* Dilation of scale 2 about (0,2)
* Rotation by 90° about (0,-2)

For rotating a polygon -90 degrees about the origin, complete the following steps:

* Dilation of scale 2 about (-3,3)
* Rotation by -90° about (3,-3)
* Dilation of scale 1/2 about (-3,3)

For reflecting a polygon across the Y-axis, complete the following steps:

* Dilation of scale 2 about (0,0)
* Reflection over the line from (2.5, 2.5) to (2.5,-1.5)
* Dilation of scale 1/2 about (0,0)
* Reflection over the line from (2.5, -1.5) to (2.5,2.5)
* Dilation of scale 2 about (0,0)
* Reflection over the line from (2.5, 2.5) to (2.5,-1.5)
* Dilation of scale 1/2 about (0,0)

For translating a T-shape across the Y-axis, complete the following steps:

* Reflection over the line from (-3.5,0.5) to (-0.5,3.5)
* Rotation by -90° about (0,0)
* Reflection over the line from (-0.5,3.5) to (-3.5,0.5)
* Rotation by -90° about (0,0)

For translating a pyramid diagonally, complete the following steps:

* Dilation of scale 1/2 about (2.5,2.5)
* Reflection over the line from (1.5,-1.5) to (-1.5,1.5)
* Dilation of scale 2 about (2.5,2.5)
* Reflection over the line from (-1.5,1.5) to (1.5,-1.5)

For positioning a square, complete the following steps:

* Dilation of scale 2 about (-4,-4)
* Reflection over the line from (2.5,-5.5) to (5.5,-2.5)
* Rotation by -90° about (-4,4)
* Reflection over the line from (5.5,-2.5) to (2.5,-5.5)
* Rotation by -90° about (-4,4)
* Dilation of scale 1/2 about (-4,-4)
* Rotation by 90° about (-4,4)

## Transformation Puzzle 3

For reflecting the star across the Y-axis, complete the following steps:

* Reflection over the line from (1.5,1.5) to (-1.5,-1.5)
* Dilation of scale 2 about (-5,0)
* Rotation by 90° about (5,0)
* Dilation of scale 1/2 about (-5,0)

For reflecting the concave polygon across the Y-axis, complete the following steps:

* Dilation of scale 2 about (0,0)
* Reflection over the line from (-2,3) to (2,3)
* Rotation by -180° about (0,2.5)
* Reflection over the line from (2,3) to (-2,3)
* Rotation by -180° about (0,2.5)
* Dilation of scale 1/2 about (0,0)
* Rotation by -180° about (0,2.5)
* Reflection over the line from (-2,3) to (2,3)

For rotating a pentagon, complete the following steps:

* Dilation of scale 2 about (5,5)
* Reflection over the line from (-1,-4) to (-4,-1)
* Rotation by 90° about (-5,-5)
* Reflection over the line from (-1,-4) to (-4,-1)
* Dilation of scale 1/2 about (5,5)

For translating a pentagon 10 units in the positive X direction, complete the following steps:

* Rotation by 90° about (0,0)
* Dilation of scale 4 about (0,-3)
* Rotation by 180° about (0,0)
* Reflection over the line from (-2,4) to (2,4)
* Dilation of scale 1/4 about (0,-3) (note: dilate by 1/2 once, click to another transformation, then dilate by 1/2 again)
* Rotation by -180° about (0,0)
* Reflection over the line from (2,4) to (-2,4)
* Rotation by -90° about (0,0)

For rotating 2 triangles across the Y-axis, complete the following steps:

* Reflection over the line from (0.5,0.5) to (3.5,3.5)
* Rotation by 90° about (0,0)

For rotating the concave polygon, complete the following steps:

* Rotation by -180° about (0,0)
* Dilation of scale 1/8 about (0,2.5)
* Rotation by -180° about (0,0)
* Reflection over the line from (2,2.5) to (-2,2.5)
* Dilation of scale 2 about (0,2.5)
* Reflection over the line from (-2,2.5) to (2,2.5)
* Rotation by -180° about (0,0)
* Dilation of scale 2 about (0,2.5)
* Reflection over the line from (2,2.5) to (-2,2.5)
* Rotation by -180° about (0,0)
* Dilation of scale 2 about (0,2.5)
* Reflection over the line from (-2,2.5) to (2,2.5)
* Rotation by -180° about (0,0)