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From Wikipedia

Balance puzzle

A number of logic puzzles exist that are based on the balancing of similar-looking items, often coins, to determine which one is of a different value within a limited number of uses of the balance scales. These differ from puzzles where items are assigned weights, in that only the relative mass of these items is relevant.


A well-known example has nine (or fewer) items, say coins (or balls), that are identical in weight save for one, which in this example we will say is lighter than the others—a counterfeit (an oddball). The difference is only perceptible by using a pair of scales balance, but only the coins themselves can be weighed, and it can only be used twice in total.

Is it possible to isolate the counterfeit coin with only two weighings?


To find a solution to the problem we first consider the maximum number of items from which one can find the lighter one in just one weighing. The maximum number possible is three. To find the lighter one we can compare any two coins, leaving the third unweighed. If the two coins tested weigh the same, then the lighter coin must be the one not on the balance - otherwise it is the one indicated as lighter by the balance.

Now, assume we have three coins wrapped in a bigger coin-shaped box. In one move, we can find which of the three boxes is lighter (this box would contain the lighter coin) and, in the second weighing, as was shown above, we can find which of the three coins within the box is lighter. So in two weighings we can find a single light coin from a set of 3*3 = 9.

Note that we could reason along the same line, further, to see that in three weighings one can find the odd-lighter coin among 27 coins and in 4 weighings, from 81 coins.

The twelve-coin problem

A more complex version exists where there are twelve coins, eleven of which are identical and one of which is different, but it is not known whether it is heavier or lighter than the others. This time the balance may be used three times to isolate the unique coin and determine its weight relative to the others.


The procedure is less straightforward for this problem, and the second and third weighings depend on what has happened previously, although that need not be the case (see below).

  • Four coins are put on each side. There are two possibilities:
1. One side is heavier than the other. If this is the case, remove three coins from the heavier side, move three coins from the lighter side to the heavier side, and place three coins that were not weighed the first time on the lighter side. (Remember which coins are which.) There are three possibilities:
1.a) The same side that was heavier the first time is still heavier. This means that either the coin that stayed there is heavier or that the coin that stayed on the lighter side is lighter. Balancing one of these against one of the other ten coins will reveal which of these is true, thus solving the puzzle.
1.b) The side that was heavier the first time is lighter the second time. This means that one of the three coins that went from the lighter side to the heavier side is the light coin. For the third attempt, weigh two of these coins against each other: if one is lighter, it is the unique coin; if they balance, the third coin is the light one.
1.c) Both sides are even. This means the one of the three coins that was removed from the heavier side is the heavy coin. For the third attempt, weigh two of these coins against each other: if one is heavier, it is the unique coin; if they balance, the third coin is the heavy one.
2. Both sides are even. If this is the case, all eight coins are identical and can be set aside. Take the four remaining coins and place three on one side of the balance. Place 3 of the 8 identical coins on the other side. There are three possibilities:
2.a) The three remaining coins are lighter. In this case you now know that one of those three coins is the odd one out and that it is lighter. Take two of those three coins and weigh them against each other. If the balance tips then the lighter coin is the odd one out. If the two coins balance then the third coin not on the balance is the odd one out and it is lighter.
2.b) The three remaining coins are heavier. In this case you now know that one of those three coins is the odd one out and that it is heavier. Take two of those three coins and weigh them against each other. If the balance tips then the heavier coin is the odd one out. If the two coins balance then the third coin not on the balance is the odd one out and it is heavier.
2.c) The three remaining coins balance. In this case you know that the unweighed coin is the odd one out. Weigh the remaining coin against one of the other 11 coins and this will tell you whether it is heavier or lighter.

With some outside the box thinking, such as assuming that there are authentic (genuine) coins at hand, a solution may be found quicker. In fact if there is one authentic coin for reference then the suspect coins can be thirteen. Number the coins from 1 to 13 and the authentic coin number 0 and perform these weightings in any order:

  • 0,1,4,5,6 against 7,10,11,12,13
  • 0,2,4,10,11 against 5,8,9,12,13
  • 0,3,8,10,12 against 6,7,9,11,13

If only one weighting is off balance then it must be one of the coins 1,2,3 which only appear in one weighting. If all weightings are off balance then it is one of the coins 10-13 that appear in all weightings. Picking out the one counterfeit coin corresponding to each of the 27 outcomes is always possible (13 coins one either too heavy or too light is 26 possibilities) except when all weightings are ballanced, in which case there is no counterfeit coin (or its weight is correct). If coins 0 and 13 are deleted from these weightings they give one generic solution to the 12-coin problem.

In literature

Niobe, the protagonist of Piers Anthony's novelWith a Tangled Skein, must solve the twelve-coin variation of this puzzle to find her son inHell: Satan has disguised the son to look identical to eleven other demons, and he is heavier or lighter depending on whether he is cursed to lie or able to speak truthfully. The solution in the book follows the given example 1.c.

Lorentz force

In physics, the Lorentz force is theforce on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:

\mathbf{F} = q[\mathbf{E} + (\mathbf{v} \times \mathbf{B})],


F is the force (in newtons)
E is the electric field (in volts per metre)
B is the magnetic field (in teslas)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in metres per second)
× is the vector cross product

or equivalently the following equation in terms of the vector potential and scalar potential:

\mathbf{F} = q \left( - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t } + \mathbf{v} \times (\nabla \times \mathbf{A})\right),


∇and∇ ×aregradient and curl, respectively
A and Φ are the magnetic vector potential and electrostatic potential, respectively, which are related to E and B by
\mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }
\mathbf{B} = \nabla \times \mathbf{A}.

Note that these are vector equations: All the quantities written in boldface are vectors (in particular, F, E, v, B, A).

The Lorentz force law has a close relationship with Faraday's law of induction.

A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail, if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).

The term qE' is called the electric force, while the term qv' ×B is called the magnetic force. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force:

\mathbf{F}_{mag} = q(\mathbf{v} \times \mathbf{B})

with the total electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.

The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force. The magnitude of this magnetic force is q v B sin θ and direction is perpendicular to the plane formed by v and B. If the particle moves perpendicular to the field, the magnitude becomes q v B and the trajectory of the particle will be circular. Also the force is in the direction perpendicular to the velocity, so magnitude of velocity will not change, i.e. the motion will be uniform circular motion.


Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, and electrically charged objects, by Henry Cavendish in 1762, obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true. Soon after the discovery in 1820 by H. C. Ørsted that a magnetic needle is acted on by a voltaic current, André-Marie Ampère that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements. In all these description, the force was always given in terms of the properties of the objects involved and the distances between them rather than in terms of electric and magnetic fields.

The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell. Interestingly Maxwell provided the equation for the Lorentz force in relation to electric cur

From Yahoo Answers

Question:On a science forces test, a question was 'if you hit a can of beans with a hammer, is it a balanced force or unbalanced force?' that was all the information they gave you. which one is it?

Answers:It is an unbalanced force...consider this definition of a balanced force... Two forces that are equal in size and opposite in direction. Balanced forces have a net force of zero. Two forces exert the same amount of force on an object, causing no change in motion. Two equal forces give off the same amount of force on an object. . When 2 objects are balanced there is no motion. So when you hit the can of beans with a hammer, there is nothing on the 'other side' of the can of beans to balance that force so it is considered an unbalanced force. I hope this helps.

Question:So I'm a college student struggling to pay for college. Its hard on my rents to pay and I do not want to make them struggle anymore. I was thinking of joining the Air Force so I could go to college while enlisted in active duty. Is four years the shortest time i can serve? Does it take sometime in there before you can start going to a surrounding college? Can I go to school full-time? How many courses does one have to take per semester? And realistically how long do you think it would take for me to get a degree? I have 53 college credits already.

Answers:i was in the same situation before i joined the air force. but it does take some time before you can start taking classes. you have to go to basic for 8 weeks then tech school from 6 weeks to a year depending on your job and then you have books you have to study for your job which you have to test on before you are allowed to take classes. i joined in july of 2008 and THIS semester which started in january is the first semester i was allowed to start taking classes. plus, the air force only likes you to take 1 or 2 classes at a time. i am taking 4 classes right now and i had to get a lot of paperwork signed by my flight chief saying i was able to take them. if you want to join for more than the education benefits then go for it. if you want to join only for the education benefits, i would look into other places to work which also pay for some of your school. have you thought of maybe working at the university where you go? a lot of colleges will give you a cut on tuition if you work somewhere on campus for them as a teachers assistant, secretary, etc.

Question:Newton's first law states that a body at rest or in uniform velocity will remain in same state of motion unless an external unbalanced force is acting on it but some force is definitely making the body move in the first place . If so isn;t that unbalanced??? Thanks in advance.

Answers:No. Uniform motion is a condition of balanced forces. Unbalanced forces result in acceleration. An object that is accelerating is not in uniform motion. An object in uniform motion most likely experienced unbalanced forces that accelerated it to its present velocity, however, the forces are now in balance. You should have said "some force *has* definitely *made* the body move in the first place". The unbalanced condition no longer exists which explains why the object is in *uniform* motion. Torg


Answers:Balanced swings left and balanced swings right

From Youtube

Balanced Forces :Balanced Forces: When the net force of an object is 0 N!

Balanced and Unbalanced Forces :Check us out at www.tutorvista.com A force is a push or a pull. A force can give energy to an object causing the object to start moving, stop moving, or change its motion. Forces occur in pairs and can be either balanced or unbalanced. Balanced forces do not cause a change in motion. They are equal in size and opposite in direction. Have you ever had an arm wrestling competition with someone? If you compete against someone who is just about as strong as you are, there will probably be a time when both of you are pushing as hard as you can, but your arms stay in the same place. This is an example of balanced forces. The force exerted by each person is equal, but they are pushing in opposite directions, in this case together. It would look something like this. Because the force that each of you is exerting is equal, the two forces cancel each other out and the resulting force is zero. Therefore, there is no change in motion. Another great place to see balanced forces in action is in a tug of war. This is similar to an arm wrestling match, only in this case the forces are moving away from each other. Just like in arm wrestling, if the two teams have equal strength, or force, the rope will stay pretty much in the same place. It would look like this. Again the resulting force is zero and there is no change in motion. Unlike balanced forces, unbalanced forces always cause a change in motion. They are not equal and opposite. When two unbalanced forces are exerted in opposite ...