Solution:

3^2 = 4 + 5
(^ is used for squaring the term, it is not used here as a mathematical symbol.)

Solution:

512

Explanation:
The cube that remains will be 8 x 8 x 8. This will make a total of 512. This is because, when a layer is taken, it will be taken from all sides of the cube which will reduce the dimension be two and not one.

Solution:

10992

Explanation:
Each girl has seven bags with seven adult cats each.
Also, each adult cat has seven little cats.
Thus each girl has 49 adult cats and 343 little cats.

Hence, seven girls will have
343 adult cats and 2410 little cats.

If we calculate the number of legs now.
Girls = 7 * 2 = 14
Cats + Little cats = 2744 * 4 = 10976

Total legs = 10976 + 14 = 10990

Since I also walked in the room, there will be two more legs.
Now total legs = 10990 + 2 = 10992.

Solution:

Solution is below

Solution:

29 rungs.
Since he is standing on the middle rung in the beginning,
Let the middle rung be x
He climbs 6 rungs first and then climbs down rungs and again climbs 18 rungs to reach the top.
Thus the position is now,
X+6-10+18 = x + 14.
This position is on the top rung.
Thus the ladder has 14 rungs above the middle rung and also it must be having 14 rungs down the middle point as well.
14+14+1 = 29.
Thus there are total 29 rungs in the ladder.

Solution:

When the boats meet for the first time, they have sailed a combined distance that is equal to one length of the lake. When they meet the second time, they have sailed 3 lengths. The elapsed time and the distance for each is three times.

When they meet for the second time, the boat M has sailed 500 x 3 = 1500 yards. Now, this is 300 yards longer than the length of the lake, it must be 1200 yards wide.

The ration between the speed of boat M and boat N is equal to the ratio of the distance that they have sailed before they meet the first time.

Solution:

Let us solve this problem with variables to find the general nature. Let us assume that there are x black chocolates and y white chocolates. Three things can happen when you take out two chocolates.
1) You take out two black chocolates.
Then, you will be replacing one black chocolate and so the jar will have x+1 black chocolates and y-2 white chocolates.
2) You take out two white chocolates.
Then, you will be replacing 1 black chocolate and so the jar will have x-1 black chocolates and y white chocolates.
3) You take out 1 black chocolate and 1 white chocolate. Then, you will be replacing a white chocolate and so the jar will have x-1 black chocolates and y white chocolates.
Thus at every step, you are wither removing a black chocolate or replacing two white chocolates by one black chocolate irreversibly till there is just one chocolate left. Therefore each white chocolate is equivalent to two black chocolates.

This, if the number of white chocolates are odd in number, the last one remaining will be white chocolate and if the number of white chocolates are even in number, the last one remaining will be black chocolate.

Solution:

No

Explanation:
7 is the only prime followed by a cube.
Assume that n3-1 is a prime for some n.
N3-1 will be equal to (n−1)(n2+n+1)
Now, n-1 divides n3-1.
If n-1>1, then we will have a contradiction to n3-1 being a prime.

Solution:

Yes We can do it.

9^(9-9)

Solution:

Let us assume that there are m male citizens
Then, the number of females will be (3553 – m)
According to question,
m/9 people received $45
And (3553-m)/12 received $60

Thus, total money spent by the wealthy man = (45 * (m / 9)) + (60 * (3552 - m) / 12)
= 5 * 3552
= $17,760