Rubick's Cube and switch Puzzles.

Consider the moves of the Rubick's cube as follows.

R -- Flip the right face in clockwise direction
Ri -- Flip the right face in anticlockwise direction

D -- Flip the down face in clockwise direction
Di -- Flip the down face in anticlockwise direction

F -- Flip the front face in clockwise direction
Fi -- Flip the front face in anticlockwise direction

U -- Flip the up face in clockwise direction
Ui -- Flip the up face in anticlockwise direction

Now consider the topmost, rightmost, and frontmost piece call it piece FTR.
I will do following moves

Ri Di R D - 6 times
R U Ri U R U U Ri - 6 times
U R Ui Li U Ri Ui L - 3 times

What will be the final position of the piece FTR after everything is done.

Four switches can be turned on or off. One is the light switch for the incandescent overhead light in the next room, which is initially off, but you don't know which. The other three switches do nothing. From the room with the switches in it, you can't see whether the light in the next room is turned on or off. You may flip the switches as often and as many times as you like, but once you enter the next room to check on the light, you must be able to say which switch controls the light without flipping the switches any further. (And you can't open the door without entering, either!) How can you determine which switch controls the light?

No retrace and 3 D to 2 D without bridge

Two more puzzles.

I will give you a set of points on a plane and also define the connectivity of all points which are connected with straight lines. There will be certain figures which can not be drawn without lifting the pen or without retracing the path. For example – You can not draw A without lifting the pen or without retracing any part of any line. You have to device a simple algorithm check which should tell this about any complicated figure.

What will the algorithm be?

Hint: While thinking about the solution – think about the counter examples.

Imagine a cube made of wires. This can be represented on a 2-D as given in the figure which follows.

I bridge is defined as in the figure which follows next.

If I connect two opposite points of the cube with a wire, will it be possible to draw that structure on 2D without using a bridge?

Hint: Think this problem as a current resistance problem.

Key Coding

Here are two -

Represent 1 as a summation of repetitive decimals containing either 0 or seven.
Hint : One of them is Bar [0.777777] -- where Bar means the sequence 777777 is repeated indefinitely.

Here is another :

Suppose you have n keys on a circular key chain and wish to put a colored sleeve on each key so that it is identifiable by its color alone, without reference to its shape. For some n this can be done with fewer than n colors. For example, if you have 4 keys then you can place the colors on the keys as follows:

Red
Red Green
Blue

The top key is the red key that is across from the blue key; and the leftmost key is the red key that is adjacent to a blue one; the other two are identifiable by their colors.
An identification scheme must work even if the key chain is flipped over, so one cannot use the words right and left. Let f(n) be the least number of colors necessary so that each key on a chain of n keys can be uniquely identified as explained above. Then f(1) = 1, f(2) = 2,
and f(3) = 3. What is f(123)?

Board to Reverse-Board

Imagine a chess board with 64 squares.

Define a Board :

1	2	3	4	5	6	7	8
9	10	11	12	13	14	15	16
17	18	19	20	21	22	23	24
25	26	27	28	29	30	31	32
33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48
49	50	51	52	53	54	55	56
57	58	59	60	61	62	63	64

Define a Reverse-Board

64	63	62	61	60	59	58	57
56	55	54	53	52	51	50	49
48	47	46	45	44	43	42	41
40	39	38	37	36	35	34	33
32	31	30	29	28	27	26	25
24	23	22	21	20	19	18	17
16	15	14	13	12	11	10	9
8	7	6	5	4	3	2	1

A 'Move' is defined as a horizontal or vertical swap between adjacent cells on the board.

The puzzle is -- How many minimum Moves are required to transform Board to Reverse-Board.