Maze Craze
Can you find a path through the factory work cells shown below, such that you alternately cross blue and red safety-light curtains as you complete the journey? You may pass through the same safety-light curtain more than once, but only as you continue to alternate the red/blue pattern.
[inset]
Light Curtains
Did you know that AutomationDirect offers safety light curtains from trusted brands like Contrinex,
Datalogic, Datasensing, and ReeR? See the complete offering of body-, hand-, and
finger-protection curtains, along with muting arms, mirrors, and other accessories here: https://go2adc.com/lightcurtains
[Image source: https://cdn.automationdirect.com/images/products/large/l_s3b.jpg]
Segment Torment
Find the length of the red line segment (x), given the other line lengths.
Prism Schism
Determine the area of the square at the top of this figure. Note that all the other bounded areas are rhombuses, save the five isosceles triangles at the base.
Credits:
Maze Craze: https://braineaser.com/brainteasers/blue-and-red-maze
Segment Torment: various attributions; see https://en.wikipedia.org/wiki/British_flag_theorem for related reading
Prism Schism: https://pororocca.com/problem/1101/
Answers below
↓↓↓↓↓↓↓
Answers:
Maze Craze:
Segment Torment:
For simplicity, lets copy the segment “x” and “5” to the right of the puzzle and add some labels, and baselines
The four right triangles allow us to set up four equalities based on the Pythagorean Theorem:
x2=a2+b2
62=c2+d2
52=a2+d2
42=c2+b2
Then add the first two and the last two equations together:
x2+62 = a2+b2+c2+d2
52+42 = a2+d2+c2+b2
Note that the right sides of these two equations are equal; therefore, the left sides must also be equal:
x2+62= 52+42
x2= 52+42-62
x2=5
x=√5
There is actually a theorem that applies directly to the general problems like this one: The British Flag Theorem (https://en.wikipedia.org/wiki/British_flag_theorem):
If a point P is chosen inside a rectangle ABCD then the sum of the squares of the distances from P to two opposite
corners of the rectangle equals the sum of the distances to the other two opposite corners. As an equation:
AP2 +CP2 = BP2 + DP2
This theorem also applies to points outside the rectangle.
Prism Schism
Note that because all the diamond shapes under the square are rhombuses, their sides are all the same length, and because the opposite sides of a rhombus are parallel, then in the following diagram, the segments labeled S1 are all parallel and of equal length. The same is true for the segments S2
This means that we can remove some “noise” from the puzzle:
And we have this simplified figure:
Let’s draw some “height” lines for the isosceles triangles:
And note that with the 90 degree corner of the square and the 180 degrees across the baseline, we can infer that angles x and y are complementary (they add up to 90 degrees).
And since the hypotenuses are the same length for the two triangles, they are effectively the same triangle, just with different orientations. So, we also know their heights:
The Area of the square = a = h2
And: h2 = 62 + 82
So, a = 100cm2