Math starts from the simple and goes a long way.
In this post, we will start from “pairing up” like 1 + 4 = 2 + 3 = 5 but there is a long way to go so that we learn the sets, working on the sets, pairing-up, observing and developing conditions for a perfect pair-up.
Pairing up for a Perfect Match
Questions chosen from previous Gauss contests
Gauss contests are organized by the Centre of Education for
Math and Computing, University of Waterloo
Problem 1
In the addition shown, P and Q each represent single digits, and the sum is 1PP7. What is P + Q?
(A) 9 (B) 12 (C) 14 (D) 15 (E) 13
Problem 2
In the right-angled triangle PQR, we have that PQ = QR. The three segments QS, TU and VW are perpendicular to PR, and the segments ST and UV are perpendicular to QR, as shown. What fraction of triangle PQR is shaded?
(A) 3 ⁄ 16 (B) 3 ⁄ 8 (C) 5 ⁄ 16 (D) 5 ⁄ 32 (E) 7 ⁄ 32
Problem 3
A box contains a total of 400 tickets that come in five colors: blue, green, red, yellow, and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of the same colour have been selected?
(A) 50 (B) 246 (C) 148 (D) 196 (E) 115
Problem 4
Greg, Charlize, and Azarah run at different but constant speeds. Each pair ran a race on a track that measured 100 m from start to finish. In the first race, when Azarah crossed the finish line, Charlize was 20 m behind. In the second race, when Charlize crossed the finish line, Greg was 10 m behind. In the third race, when Azarah crossed the finish line, how many metres was Greg behind?
(A) 20 (B) 25 (C) 28 (D) 32 (E) 40
Problem 5
In right-angled, isosceles triangle FGH, segment FH = √̅8. Arc FH is part of the circumference of a circle with centre G and radius GH. The area of the shaded region is
(A) π – 2; (B) 4 π – 2 (C) 4 π – (1 ⁄ 2) √̅8 ; (D) 4 π – 4 (E) π – √̅8
About a decade ago, there was a great Tsunami happening in Asia.
What do you know about the Asian tsunami?
Read through the article first. Use the following numbers to fill in the blanks in the story.Think about which numbers make sense.
500 20 8,000; 2004 110,000 30,000 9.0
A tsunami triggered by a very large earthquake off the coast of the
Indonesian island of Sumatra on December 26, ____, has left
more than 150,000 people dead and millions homeless. Countries hit hardest by the disaster include
Sri Lanka, Indonesia, India, Thailand, and the Maldives. Almost 75% of the deaths occurred in
Indonesia, estimated at ____. Sir Lanka was second highest with about 20% of the estimated deaths, or
______ people lost that day. The rest of the deaths, approximately ____, occurred in the other nine
countries affected by the tsunami.
The ____ foot wall of water, higher than a two-story building,
swallowed entire villages. The tsunami waves were not only very high, they moved at a much faster speed
than normal. These waves were comparable in size to those you see on some of the surfing movies;
but those waves travel at 30 miles an hour, and the tsunami waves
were moving more than fifteen times as fast at ____ miles an hour.
The velocity of the force is what caused the destruction—a massive force that swept away everything in its path.
The earthquake causing this Tsunami was a destructive earthquake measuring ______ on the Richter scale,
the fourth worst earthquake in recorded history. Earthquakes are measured on a Richter scale that has
a range from 0 to 12; a 6.0 on the scale is a pretty bad earthquake.
(Story constructed from January 2005 news reports)
Where does the symbol π come from?
In 1652, William Oughtred used π to refer to the periphery of a circle (in his expression, the ratio of circumference-to-diameter of a circle is π ⁄ δ, the latter referring to diameter).
In 1665, Jonh Wallis used a Hebrew letter 
(mem) to equal the ratio of one-quarter of circumference to diameter of a circle. (This letter plays the role as of “M” in Latin alphabets, but look how close its shape resembles a quarter of a circle, as well as the Greek letter pi !)
In 1705, William Johns used π to represent the ratio of circumference-to-diameter of a circle (believed to be first use with exactly same meaning as in today) .
From 1736, Leonard Euler, both famous and a prolific writer in mathematics works, spread the use of π in his publications.
Counting from the first relevant use, the symbol π has already had a history of more than 360 years!
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)
Prime numbers are those that have 1 (one) and itself as the only two divisors. Examples of primes are 2, 3, 5, 7, 11. None of 4, 6, 9 is a prime since 4 = 2 × 2, 6 = 2 × 3, and 9 = 3 × 3.
If a number greater than one is not a prime, then it is a composite number, and can be factored into the product of primes — called prime factorization. We have given the prime factorization of 4, 6, 9 as above. For a couple of more examples:
12 = 2 × 2 × 3
36 = 2 × 3 × 3 × 3
28 = 2 × 2 × 7
So all natural numbers are divided into three classes: the number 1, the prime numbers, and the composite numbers.
A bonus point: π, besides representing in a circle, the ratio of circumference to diameter, also stands for a special function related to prime numbers. Function π(x) — for every integer x, represent the number of primes less than or equal (i.e. not exceeding) x. For example, we have:
π(2) = 1, π(3) = 2, π(10) = 4, π(20) = 8 etc.
[To find why π(10) = 4, recall the 4 prime numbers not exceeding 10: they are 2,3,5, and 7.]
★ On the Patterns
Number Patterns * Numbers, Colors and Stars
Taking a look at the following chart.
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1 |
2 |
★3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
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11 |
12 |
13 |
★14 |
15 |
16 |
17 |
18 |
19 |
20 |
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21 |
22 |
23 |
24 |
★25 |
26 |
27 |
28 |
29 |
30 |
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31 |
32 |
33 |
34 |
35 |
★36 |
37 |
38 |
39 |
40 |
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41 |
42 |
43 |
44 |
45 |
46 |
★47 |
48 |
49 |
50 |
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51 |
52 |
53 |
54 |
55 |
56 |
57 |
★58 |
59 |
60 |
The following questions are for everyone! Whether you are students in elementary schools (grade 3 and up), or in middle schools; or just parents who are tutoring your child; here are the challenges!
First, let us look up to the stars:
(1) What pattern is followed for the placement of stars ?
(2) Each number besides the star is increased by ____. (Fill in the blank)
(3) Following this pattern, the three more numbers that are besides and that comes after 58 are: __ , __ , and __.
(4) Now go back to the chart, and let us look at the colored cell (those colored by yellow)
(5) What can you do to follow the yellowed-colored cell? Please color the numbers on the chart by continue the pattern that you discovered.
(6) What are the common features of these yellow cells? Color some new cells, and explain how, by coloring these cells, you have followed and extended the pattern which is already in the chart.
Take a moment to think. You can answer these questions!
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?
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.
With two set of restrictions on what kind of tools you can use, there are actually two questions. 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).
See the following article on how to do it in general.
How to find the center of a given circle?
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. This will be discussed in another posting.