Doesn’t fit? Reconstruct!
Sprouts Game
Equi-extended and isoperimetric non-congruent triangles
The picture below shows the ONLY one pair of triangles with the following properties:
· One triangle is a right triangle and one is isosceles,
· All side lengths of both triangles are rational numbers, and
· The perimeters and areas of both triangles are equal.
Icosahedron with golden ratio cross-sections
3 intersecting golden rectangles (1 : φ) will create the vertices of an icosahedron.
Equable Triangles
There are only five integer-sided triangles whose area is numerically equal to its perimeter:
(5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17)
As you can see from the picture, only 2 of them are right triangles.
Elementary 4-manifold topology
Cut any of the loops of strings below and the whole chain comes apart.
You will find many of these kinds of puzzles in my book “Impossible Folding Puzzles and Other Mathematical Paradoxes” .
Infinite Pythagorean Triplets
Consider the following simple progression of whole and fractional numbers (with odd denominators):
1 1/3, 2 2/5, 3 3/7, 4 4/9, 5 5/11, 6 6/13, 7 7/15, 8 8/17, 9 9/19, …
Any term of this progression can produce a Pythagorean triplet, for instance:
4 4/9 = 40/9; the numbers 40 and 9 are the sides of a right triangle, and the hypotenuse is one greater than the largest side (40 + 1 = 41).
Golden Ratio (And Its Inverse) In Yin Yang
The philosophy of the Yin Yang is depicted by the the “taichi symbol” (taijitu). In fact, Yin Yang is a concept of dualism, describing how seemingly opposite or contrary forces may actually be complementary,
Curiously enough, in the taichi symbol are hidden the golden ratio and its reverse. As shown in the picture. Continue reading “Golden Ratio (And Its Inverse) In Yin Yang”
Target 10
Here is a little puzzle of our creation you can make with your kids or in class…
“Stubborn” Number 33
It is conjectured that n is a sum of 3 cubes if n is a number that is not congruent to 4 or 5 mod 9. The number 33 enters this category, but for 64 years no solutions emerged — that is, whether the equation 33 = x³ + y³ + z³ has an integer solution. Continue reading ““Stubborn” Number 33″