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Uncle Shiva has gifted Nikhil a new toy. Uncle Shiva believes in gifting toys with a mathematical flavour and this is no different. This toy consists of a wooden board with wooden pegs. The pegs are arranged in N columns with 3 pegs in each column. Each peg is coloured Red, Blue or Green.
Nikhil has a set of N − 1 elastic bands (i.e. rubber bands). He builds a chain of bands linking column 1 to column N as follows. He starts by placing a band from a peg in column 1 to a peg in column 2. From the peg where the first band ends, he places a second band connecting that peg in column 2 to a peg in column 3. Continuing in this way, he places all N − 1 bands to connect the N columns.
While building this chain of elastic bands, he is not allowed to connect two pegs at the same position in adjacent columns. So for instance, the second peg on column i cannot be connected to the second peg on column i − 1 or the second peg on column i + 1. It may be connected to any other peg in those two neighbouring columns.
Uncle Shiva has added a constraint. Nikhil has to ar...
The King of Zyorg has a cabinet with N ministers and they have assembled for a meeting. The King enters the meeting chamber to find that the ministers have arranged themselves so that friends sit with friends.
The seats are numbered 1 . . . N and we refer to the minister sitting on the i th seat in this initial arrangment as i. So, the initial arrangment is of the ministers is (1, 2, 3 . . . N).
The King orders the ministers to rearrange themselves and finds that some of the friends have just exchanged seats. The enraged King then orders that they must rearrange themselves so that for every k < N the set of ministers sitting in the first k chairs DOES NOT consist only of the ministers 1, 2, . . . k.
He then wonders if this is possible at all and quickly convinces himself that this is so. Then he wonders how many different ways can they rearrange themselves fulfilling his order.
For example, suppose N = 3. The initial arrangment is of course (1, 2, 3). If they rearrange themselves as (2, 1, 3) this ...
The King of Zyorg has a cabinet with N ministers and they have assembled for a meeting. The King enters the meeting chamber to find that the ministers have arranged themselves so that friends sit with friends.
The seats are numbered 1 . . . N and we refer to the minister sitting on the i th seat in this initial arrangment as i. So, the initial arrangment is of the ministers is (1, 2, 3 . . . N).
The King orders the ministers to rearrange themselves and finds that some of the friends have just exchanged seats. The enraged King then orders that they must rearrange themselves so that for every k < N the set of ministers sitting in the first k chairs DOES NOT consist only of the ministers 1, 2, . . . k.
He then wonders if this is possible at all and quickly convinces himself that this is so. Then he wonders how many different ways can they rearrange themselves fulfilling his order.
For example, suppose N = 3. The initial arrangment is of course (1, 2, 3). If they rearrange themselves as (2, 1, 3) this ...