Tag: G 8-10

Categories
Geometry /Shapes

Find the Center of a Circle — Do you know how to do it?

Using a compass, you can draw a circle at any place, with any radius.

Now let’s reverse the problem. Given a circle, do you know how to find its center? (Of course, once the circle is found, there shall be no problem at all to tell its diameter, or radius.) You only see the circle itself, there is no explicit indication on where the center is.

In general, you can find the center using any convenient method, including copy-and-paste the circle onto a paper, and then fold it.

In particular (from classical Euclidean geometry), where it’s required to do so with a ruler (with which you are allowed to draw lines and line segments only) and a compass (with which you are allowed to draw circles only).

If you attempt to solve this problem with a ruler and a compass, then you are required to know how to make a perpendicular bisector. Do you know how to do it?

Categories
Math for Early Ages (G7 and under) Math Modelling

cut a pizza into 11 equal slices, exactly evenly

What is the easiest way to cut a pizza into 11 equal slices?

The step-by-step solution is:

  • Take a wrist watch.
  • Position the watch hands to noon (12:00 exact), and put in the center of pizza.
  • Cut in the direction of watch hands.
  • Advance the watch until next overlapping of its hands…
Categories
Geometry /Shapes

3D objects with 3 views from top, front and side

For a 3D objects, given three views to you: one from top, one from front, and one from side, can you imagine what the original 3D objects looks like?

The question is not posed to a mechanic engineer, it would be trivial in that case. The question is raised to get a junior middle student to think a bit.

For a cylinder one of the three views is a circle, and the other two views are rectangles. For a cone one of the views is a circle, and the other two views are triangles. What if the three views given are a circle, a rectangle and a triangle? Can you figure out the original shape?

Categories
Geometry /Shapes Math BASICS Numbers

Protected: The (3-4-5) Pattern for Pythagorean Triplet [Thinking-of sides of a right triangle]

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Categories
Algebra Numbers

Complete Numbers in Fraction Equations

The formula on our face page of “amazing numbers” is rather interesting:
1 – (1 ⁄ 28) = (1 ⁄ 2) + (1 ⁄ 4) + (1 ⁄ 7) + (1 ⁄ 14)

The point of interest is that: if you look at all divisors of 28: they are 1,2,4,7,14,28; with the exception of 28 which is itself, all divisors have appeared in this formula, and they appear in the form of so-called “unit fraction”, where numerator is 1. So (1 ⁄ 2), (1 ⁄ 4), etc. are all unit fractions.

Indeed, we present a fraction equation to make it a bit unusual, but there is a low-pitch but straightforward ways to present number 28. We have that:
28 = 1 + 2 + 4 + 7 + 14
To get to the earlier fraction form, just divide every term by the number 28.

The smallest complete number is 6 (=1+2+3), 28 is the 2nd complete number, and after that, you will not see a complete number until 496. So complete numbers are rare among all positive whole numbers.

Complete numbers 6 also has a nice fraction form, as:
1 – (1⁄6) = (1⁄2) + (1⁄3)

Categories
Numbers

What’s so special about number 8208?

The seemingly trivial, innocent number 8208 actually has something special. It is a member of a niche class of numbers called narcissistic numbers.

What’s so special?

First note it has 4 digits. If we raise every digit 8, 2, 0, 8 to the 4th power, and then add them together:

84 + 24 + 04 +84 = 8208

(To find 84, just think 84 = 212 = 4 x 210 = 4096)

Now read the four digits of the total. It’s 8, 2, 0, 8 again!

See the point? All the digits of the number are applied to generate that same number. Repeat the same procedure once more, you still get the same number. Seems this number loves itself so its ego is repeated. — That speaks for its name: narcissistic.

 

The 3-digit number 153 has similar property.

Since it has 3 digits, we raise every digit 1, 5, 3 to the 3rd power, and add them together!

13 + 53 + 33 = 153

Another time to witness the magic!

** **

Any one-digit number belongs to the class of narcissistic numbers. Reason like this: (if you raise everything to exponent one, it is just itself); this is plain yet may be too trivial to be interesting.

And tell you another secret. There is NOT ANY 2-digit number that is narcissistic (we assume that tens-digit cannot be zero: it has to be between 10 and 99). Some may be close (almost narcissistic!): for example

32 + 52 = 34;  72 + 52 = 74  (note 34 is just one off from 35)

That’s it. Happy reading!

 

Categories
Algebra

Power Mad

Continue reading “Power Mad”