Application of "Simplifying a Puzzle"

The Ant and The Cube :

An ant has to travel from A to B (which are diagonally opposite vertices of a cube) and back to A. The ant moves from A to B through a thread initially present between A and B, Once the ant reaches B the thread is removed, Now the ant has to come back to A through shortest path using the walls of the cube. If the velocity of the ant is same while going and coming then what is the ratio of time taken?





Solution 1 : (conventional method of solving)

(A-B) : distance covered = D1 = √(a^2+(√2 a)^2) = √3 a
(B-A) : distance covered = D2 = √(x^2+a^2) + √((a-x)^2+a^2)

for minimum D2, d(D2)/dx = 0

Finding d(D2)/dx will take some time....
Finally you will get the condition for minimum D2 is x=a/2
So D2= √5 a

So time taken while coming back from B to A is √(5/3) times more than the time taken while going from A to B as the speed is constant.

Solution 2 : (using the simplifying a puzzle technique)

This Problem will become lot simpler by just simplifying it with little visualization. When the ant has to come back from B to A we can bring the MNBO Face of cube on same plane as AMNP. i.e. rotating the face MNBO by 90 degree in clockwise direction. Now the Ant has to travel from B to A which is nothing but the diagonal of the rectangle APBO.

Now the distance between two points B and A is √((AP)^2+(BP)^2) = √5 a

So time taken while coming back from B to A is √(5/3) times more than the time taken while going from A to B as the speed is constant.

Solution : 7 Coin Puzzle

Pre-analysis :

To solve the puzzle, you might now start a division into cases. You notice, of course, that placing the first coin on 1 is all right, because all vertices play the same role; therefore, with regard to the first coin you do not need to make any division into cases (on 1, on 2, etc.). For the second coin, you have the seven possibilities of placing it on 2, on 3, . . . , on 8; it is irrelevant along which free line you move (when the vertex can still be occupied along two free lines, which is not possible for the vertices 4 and 6). If the second coin has also been placed, you can start to make assumptions about the third coin; it cannot always be placed in six ways, for if the second coin has been placed on 3, then the third coin cannot be placed on 6, because then neither of the lines 1-6 and 3-6 is free any more. Proceeding in this way, you will find all solutions after much sifting. If what you want is to present the puzzle to others, you can stop as soon as a solution has been found; you then note it down and learn it by heart. However, someone who proceeds thus has not seen through the puzzle, even if he finds a solution. Yet, one might say he has followed the directions about systematic trial and error. Indeed he has, and certainly there are still less efficient ways of solving the puzzle. Many a solver who looks for the solution starts pushing at random, and if, after much pushing, he has been able to place the seven coins, he has the bad luck of not remembering how he did it. This, of course, is puzzle solving of the worst kind.

Solution :

In order to understand the puzzle completely, you should observe that you ought to try to retain as many free lines as possible, and hence you should make as few lines as possible useless. Placing the first coin on 1 makes two lines useless: 1-4 and 1-6. Placing the second coin makes two more lines useless, except when this coin is placed on 4 or on 6. The correct continuation is, therefore: on 4 or on 6; for example, by occupying 4, only the line 4-7 is made useless, because 4-1 had already dropped out; the third coin should now be placed on 6 or 7, etc. The coins should always be put on adjacent vertices (such as 3, 6, 1, 4); by adjacent vertices we mean vertices that are connected by a line, like 3 and 6.

The puzzle can be simplified considerably by observing that the eight lines form an octagon, and that the shape of the octagon reduces the complexity of puzzle to be an obvious easy one.

Post-analysis :

Now the seemingly complex problem is reduced to a very simple puzzle which everyone can solve instantly. It’s a very nice and handy technique of solving puzzles but it needs a lot of visualization and imagining capabilities.

Simplifying a Puzzle

The puzzle of the 7 coins : The object is to place 7 coins at 7 vertices out of the 8 vertices of the star. When placing any coin you should move it along a free line, and put it down at the end of that line. A line is called free if there is no coin at either of its endpoints.

Suppose that we first put a coin on 1 by moving it along the line 4-1, then a coin on 2, moving along 5-2, next a coin on 3 (along 6-3), a coin on 4 (along 7-4), and a coin on 5 (along 8-5). Then we are played out, because there is no free line left; hence, we can place only five of the seven coins in this way.

Puzzles ?

One can make a deduction which is quite certainly the ultimate truth of puzzles. Despite appearances, puzzling is not a solitary game, every move the puzzler makes, the puzzle-maker has made before; every piece the puzzler picks up, and picks up again, and studies and strokes, every combination he tries, and tries a second time, every blunder and every insight, each hope and each discouragement have all been designed, calculated, and decided by the other.

For everyone solving a good puzzle gives a teasing sense out of an undifferentiated mass of symbols and images is a sublime experience.

A well framed puzzle has a non-obvious solution and requires some sort of inspired insight on the part of the solver, but once the solver has deduced—and executed—the mechanism, the answer must be unambiguously clear. In that sense a good puzzle is the opposite of a real-life problem, which demands compromises and trade-offs, and rarely provides a way to confirm the existence, much less the nature, of the "correct" solution. On the contrary every puzzle is inspired from the real life and modified to remove the controversies and to bring consensus between the master mind and the solver. Solving puzzles and learning new techniques improves the logical thinking and boosts confidence in taking up new challenges.

Try Puzzles for fun, solve them if you have it in you and analyse them if you have passion.