Product of Two Numbers generate Maximum value

Puzzle # 14

Using the digits 1 up to 9, two numbers must be made. The product of these two numbers should be as large as possible. All digits must be used exactly once

• By Multiplying only Two Numbers say A & B, Maximum Value needs to be generated.
• These two numbers A and B should cover all the digits from 1 to 9.

What are the TwoNumber (A & B) ??


Solution
Remember two rules to generate Max Product.

1. Difference of two numbers should be minimum to attain Max Product
2. Higher value digits be placed in Most significant side of Number i.e., Let us say number A is a 5 digit number and assume it's value to be A=abcde
1. In above a,b,c,d,e are digits.
2. Possible digit values are a > b > c > d > e

If we follow the above rules, Then numbers are

A = 96521

B = 8743

Note : In above while calculating the Numbers, A Snake structure is followed starting from 9 as shown below :

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True Magical Number

Puzzle # 13

A Very Magical Number(Dividend) that leaves a reminder that is one less than number(Divisor) that was used to perform the division

• When Magical Number(Dividend) divided by 2 leaves Reminder 1.
• When Magical Number(Dividend) divided by 3 leaves Reminder 2.
• When Magical Number(Dividend) divided by 4 leaves Reminder 3.
• When Magical Number(Dividend) divided by 5 leaves Reminder 4.
• When Magical Number(Dividend) divided by 6 leaves Reminder 5.
• When Magical Number(Dividend) divided by 7 leaves Reminder 6.
• When Magical Number(Dividend) divided by 8 leaves Reminder 7.
• When Magical Number(Dividend) divided by 9 leaves Reminder 8.
• When Magical Number(Dividend) divided by 10 leaves Reminder 9.

What is the Magical Number (Dividend) ??
Magical Number is not very small neither too big. Think of it!!


Solution
In all the above specified conditions, Common thing to observe is every time when Magical Number (dividend) is divided by a particular number yields a reminder that is 1 less than the particular number.

From our basic Math, we know that Least common multiple satisfying above all condition will yield a reminder Zero. and LCM for 1,2,3,4,5,6,7,8,9,10 is 

Required Magical Number = LCM -1 
                                        = 2520 - 1
                                        = 2519


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Passing on a Truck over a bridge

Puzzle # 12

There is a bridge that can withstand a total weight of Maximum 10 Tons. Above which the bridge will be collapsed !

• Bridge covers a total distance of 5KM.
• This bridge is maintained by highly authorised officials who will thoroughly check the vehicle before allowing to passed over it.
• Under any circumstance, officials are allowed Max of 10 Tons over the bridge.

A Truck weigh exactly 10 Tons(Load + Truck + Driver..) i.e it includes everything? It is allowed to cross over the bridge and after covering a distance of 4.75 KM. A crow (Max - 1.5 KG) came and sat over the truck. Luckily bridge didn't got collapsed. What could be the reason behind this? When driver didn't dropped any of his items or truck parts!!


Solution
1 TON = 1000Kgs

Truck weighs exactly 10,000 Kgs and is moving on the Bridge.As Truck travelled a distance of 4.75 Kms on bridge then it should have consumed some fuel. 
On an average truck will consume 1 liter of Diesel for every 3 Kms that it has travelled.

Truck has travelled around 4.75 Kms, then it should have reduced by weight more than 1.5 Kgs. This reduced weight will compensate the weight of the Crow that came and sat on the Truck.

Hence Truck has crossed bridge without getting collapsed!!


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Ants moving towards Corners

Puzzle # 11

There are 'n' ants at 'n' corners of a 'n' sided closed regular polygon, they randomly start moving towards another corner that is adjacent to it?

If n = 5, PENTAGON.. i.e., 5 ants positioned at 5 corners are started moving towards other possible corners
If n= 6, HEXAGON.. i.e., 6 ants positioned at 6 corners are started moving towards other possible corners
If n = 8, OCTAGON.. i.e., 8 ants positioned at 8 corners are started moving towards other possible corners
So on and So Forth.

What is the probability that they don't collide?


Solution
PROBABILITY = 1/ 2^{n - 1}

There are 'n' corners in a regular polygon.
Ant placed in 1st corner can go in 2 directions along the closed. Similarly ants placed in any corner can move  in 2 directions.

Total possible directions that ants can move in 'n' sided regular polygon is 2 x 2 x 2 ... n times.
There are only 2 possible solutions where ants cannot collide i.e,
1. CLOCKWISE 
2. ANTICLOCKWISE 

Probability that ants will not collide each other = 2 / 2ⁿ 

                                                                      = 1 / 2^{n - 1}

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Combination of pieces on Chess board

Puzzle # 10

How many ways are there to arrange the sixteen black or white pieces of a standard international chess set on the first two rows of the board?

1. Each square can hold only one chess piece
2. All eight Pawns are identical
3. Similarly each rook, knight and bishop are identical to its pair.

What could be the possible combinations to arrange the chess pieces on chess board?


Solution
Total available squares to arrange pieces on chess board= 16

       6,48,64,800 WAYS !!

There are total 16 pieces which can be arranged on 16 places ¹⁶P₁₆ = 16!

(16! = 16 * 15 * 14 * 13 * 12 * ..... * 3 * 2 * 1)but, there are some duplicate combinations because of identical pieces.

There are 8 identical pawn, which can be arranged in ⁸P₈     = 8!

There are 2 identical Bishops, which can be arranged in ²P₂ = 2!

There are 2 identical Rooks, which can be arranged in ²P₂   = 2!

There are 2 identical Knights, which can be arranged in ²P₂  = 2!

Hence, the require answer is = (16!) / (8! * 2! * 2! * 2!)
                                            = 6,48,64,800

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