(Puzzles) Challenging Mathematical Puzzles (25)

Puzzles : Challenging Mathematical Puzzles

75. Four words add up to a fifth word numerically:
mars
venus
uranus
saturn
-------- +
neptune 

Each of the ten letters (m, a, r, s, v, e, n, u, t, and p) represent a unique number from the range 0 .. 9.
Furthermore, numbers 1 and 6 are being used most frequently.

Answer

The easiest way to solve this problem is by writing a computer program that systematically tries all possible mappings from the numbers onto the letters. This will give you only one solution which meets the condition that numbers 1 and 6 are most frequently used.


mars m = 4
venus a = 5
uranus r = 9
saturn s = 3
-------- + v = 2 4593
neptune e = 0 20163

n = 1 695163
u = 6 358691
t = 8 -------- +
p = 7 1078610 

 

76. There are 4 army men. They have been captured by a rebel group and have been held at ransom. An army intelligent officer orders them to be burried deep in dirt up to their necks. The format of their burrial are as shown in the figure.

Conditions

* They each have hats on their heads. either black(b) or white (w) look at diagram above. There are total 2 white hats and 2 black hats.
* They only look in front of them not behind. They are not allowed to communicate by talking.
* Between army man 1 and 2, there is a wall.
* Captive man 4 can see the colour of hats on 2 and 3
* 3 can only see 2's hat
* 2 can only see a wall and 1 can see a wall too, but is on the other side

The officer speaks up, "If one of you can correctly tell me the colour of your hat, you will all go scott free back to your contries. If you are wrong, you will all be killed.

How can one of them be certain about the hat they are wearing and not risk the lives of their fellow souldiers by taking a 50/50 guess!


Answer : -

Either soldier 3 or soldier 4 can save the life as soldier 1 and soldier 2 can not see colour of any hat, even not their own.. In our case soldier 3 will tell the colour of his hat.

Soldier 4 can see the hat on soldier 2 and soldier 3. If both are white, then he can be sure about colour of his hat which will be black and vice-versa. But if one of them is white and one is black, then soldier 4 can not say anything as he can have either of them. So he will keep mum.

If soldier 4 won't say anyhing for a while, then soldier 3 will know that soldier 4 is not in position to tell the colour of hat on his hat. It means that colour of soldier 3's hat is opposite of colour of soldier 2's hat. So soldier 3 can tell correctly the colour of hat on his head which is Black. 

 

77. One side of the bottom layer of a triangular pyramid has 12 balls. How many are there in the whole pyramid?
Note that the pyramid is equilateral and solid.

Answer : -

There are total 364 balls.

As there are 12 balls along one side, it means that there are 12 layers of balls. The top most layer has 1 ball. The second layer has 3 (1+2) balls. The third layer has 6 (1+2+3) balls. The fourth layer has 10 (1+2+3+4) balls. The fifth layer has 15 (1+2+3+4+5) balls. Similarly, there are 21, 28, 36, 45, 55, 66 and 78 balls in the remaining layers.

Hence, the total number of balls are
= 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78
= 364 balls 

 

78. A blindfolded man is asked to sit in the front of a carrom board. The holes of the board are shut with lids in random order, i.e. any number of all the four holes can be shut or open.

Now the man is supposed to touch any two holes at a time and can do the following.

    * Open the closed hole.
    * Close the open hole.
    * Let the hole be as it is.

After he has done it, the carrom board is rotated and again brought to some position. The man is again not aware of what are the holes which are open or closed.

How many minimum number of turns does the blindfolded man require to either open all the holes or close all the holes?
Note that whenever all the holes are either open or close, there will be an alarm so that the blindfolded man will know that he has won.


Answer : -

The blindfolded man requires 5 turns.

1. Open two adjacent holes.
2. Open two diagonal holes. Now atleast 3 holes are open. If 4th hole is also open, then you are done. If not, the 4th hole is close.
3. Check two diagonal holes.
    * If one is close, open it and all the holes are open.
    * If both are close, open any one hole. Now, two holes are open and two are close. The diagonal holes are in the opposite status i.e. in both the diagonals, one hole is open and one is close.



4. Check any two adjacent holes.
    * If both are open, close both of them. Now, all holes are close.
    * If both are close, open both of them. Now, all holes are open.
    * If one is open and one is close, invert them i.e. close the open hole and open the close hole. Now, the diagonal holes are in the same status i.e. two holes in one diagonal are open and in other are close.



5. Check any two diagonal holes.
    * If both are open, close both of them. Now, all holes are close.
    * If both are close, open both of them. Now, all holes are open.

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