Caster angle - Vehicle physics

Caster angle is the angular displacement from the vertical axis of the suspension of a steered wheel in a car, bicycle or other vehicle, measured in the longitudinal direction. It is the angle between the pivot line (in a car - an imaginary line that runs through the center of the upper ball joint to the center of the lower ball joint) and vertical. Car racers sometimes adjust caster angle to optimize their car's handling characteristics in particular driving situations.
steering are angled such that a line drawn through them intersects the road surface slightly ahead of the contact point of the wheel. The purpose of this is to provide a degree of self-centering for the steering - the wheel casters around so as to trail behind the axis of steering.
Read more about this on Wikipedia (link)

Camber angle - angle made by the wheel of an automobile

If you ever thought about what happens to a car wheel when it turns, then this Wikipedia link will help you understand in greater detail.
Camber angle is the angle made by the wheel of an automobile; specifically, it is the angle between the vertical axis of the wheel and the vertical axis of the vehicle when viewed from the front or rear. It is used in the design of steering and suspension. If the top of the wheel is further out than the bottom (that is, away from the axle), it is called positive camber; if the bottom of the wheel is further out than the top, it is called negative camber.
Camber angle alters the handling qualities of a particular suspension design; in particular, negative camber improves grip when cornering. This is because it places the tire at a more optimal angle to the road, transmitting the forces through the vertical plane of the tire, rather than through a shear force across it. Another reason for negative camber is that a rubber tire tends to roll on itself while cornering. If the tire had zero camber, the inside edge of the contact patch would begin to lift off of the ground, thereby reducing the area of the contact patch.

A* search algorithm

In computer science, A* (pronounced "A star") is a best-first, graph search algorithm that finds the least-cost path from a given initial node to one goal node (out of one or more possible goals).
A* incrementally searches all routes leading from the starting point until it finds the shortest path to a goal. Like all informed search algorithms, it searches first the routes that appear to be most likely to lead towards the goal. What sets A* apart from a greedy best-first search is that it also takes the distance already traveled into account (the g(x) part of the heuristic is the cost from the start, and not simply the local cost from the previously expanded node).
Read more at this link.

Puzzle about measuring difference in weights

Q. Ali the thief had managed to break into the Sultan's treasure room. He was looking for a bag containing 1250 diamonds, each weighing 0.8 grams. Unfortunately, the Sultan had filled nine other bags with imitation diamonds. There were 1000 in each bag, each one weighing 1 gram. So all 10 bags weighed exactly 1000 grams and all the bags looked exactly alike. Just as Ali was scratching his head, the Sultan burst in with his guards. As he was a merciful Sultan, he gave Ali a chance to save his life. Here is what he said; "If you can find the bag which contains the real diamonds by using these scales just once, you may keep the diamonds. If not, you will be beheaded. You can take
stones from any of the bags and put them in other bags if you wish but you can only weigh once!" How did Ali save his neck?

Ans: Ali numbered the ten bags from 1 to 10. He then took ten stones from bag 1, twenty stones from bag 2, thirty stones from bag 3 .... likewise till hundred stones from bag 10 and weighed them together. Had all the stones weighed 1 gm in all ten bags, the total weight would have been 550 grams. since one of the bags had real diamonds of lesser weight ( by 0.2 gm each ), Ali had lesser weight than 550 gms. If Ali had the weight measured as 548 gms, bag 1 from which Ali had taken ten stones contained real diamonds. If the weight had been 546 gms, bag 2 had real diamonds etc. Thus he knew which bag contained the real diamonds by the measured weight by measuring only once.

Maths: Armstrong Numbers

Question: Divide the number 1301 into four three digit parts abc, def, ghi and jkl

so that

abc + def + ghi + jkl = 1301

and

abc = a^3 + b^3 + c^3
def = d^3 + e^3 + f^3
ghi = g^3 + h^3 + i^3
jkl = j^3 + k^3 + l^3




SOLUTION :

' This is related to “Armstrong Numbers”.

1301 = 153 + 370 + 371 + 407

These are the only 3 digit numbers that have this property.

The number 1729 (called Hardy Ramanujam number)

Why is the number 1729 famous ? Well, it all started from around a century back. There were these 2 famous mathematicians, G Hardy and Srinivasa Ramanujam (who was entirely untrained, but a remarkable mathematician) who were sitting in a hospital room (Hardy had come to visit Ramanujam).
Ramanujam asked Hardy about the taxi number that he came in, and Hardy mentioned 1729, calling it a boring number. Ramanujam disputed that,calling 1729 as a very interesting number, it is the smallest number expressible as the sum of two cubes in two different ways.
Which are the 2 ways?
10X10X10 + 9X9X9 = 1729
12X12X12 + 1X1X1 = 1729
There are no other numbers smaller than this which is the sum of 2 cubes.

The mysterious number 6174

I was roaming around the web, when I saw this particular web page, explaining the mystery of the number 6174. Read it to the end, and you will feel happy that you learnt something new.
So what's the mystery of the number 6174? Well, it's sort of like a mathematical puzzle. You need to do the following steps:
1. Take any 4 digit number where all 4 digits are not the same (so you can't select 9999), so for example, take 3456
2. Rearrange the digits to get the largest and smallest number: so you get 3456 and 6543
3. Subtract them, so you get 6543-3456 = 3087
4. Do this again with the resultant number, so you get 8730 - 378 = 8352
5. Again repeat 4, so you get 6174
6. Do it again, 7641-1467 = 6174
The number repeated itself, and will do so again and again, no matter which 4 digit number you take.
For 3 digits number, you get 495

Some maths calculations resulting in visual similarity

Sometimes maths can be real fun in terms of how some simple calculations can result in some visually interesting results. Take a look at some of the equations below, and you can't help but admire the way that these work out

Multiplication and addition with increasing numbers
0 x 9 + 0 = 0
1 x 9 + 1 = 10
12 x 9 + 2 = 110
123 x 9 + 3 = 1110
1234 x 9 + 4 = 11110
12345 x 9 + 5 = 111110
123456 x 9 + 6 = 1111110
1234567 x 9 + 7 = 11111110
12345678 x 9 + 8 = 111111110
123456789 x 9 + 9 = 1111111110


Multiplications with similar increasing numbers
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321


Another Magic of Math

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

Then... Check this out ...

1 x 18 + 1 = 19
12 x 18 + 2 = 218
123 x 18 + 3 = 2217
1234 x 18 + 4 = 22216
12345 x 18 + 5 = 222215
123456 x 18 + 6 = 2222214
1234567 x 18 + 7 = 22222213
12345678 x 18 + 8 = 222222212
123456789 x 18 + 9 = 2222222211