# (Puzzles) Challenging Mathematical Puzzles

**Puzzles : Challenging Mathematical Puzzles**

**1. Three friends divided some bullets equally. After all of them shot 4
bullets the total number of bullets remaining is equal to the bullets each had
after division. Find the original number divided.**

**Answer** **18**

Assume that initial there were 3*X bullets.

So they got X bullets each after division.

All of them shot 4 bullets. So now they have (X - 4) bullets each.

But it is given that,after they shot 4 bullets each, total number of bullets
remaining is equal to the bullets each had after division i.e. X

Therefore, the equation is

3 * (X - 4) = X

3 * X - 12 = X

2 * X = 12

X = 6

Therefore the total bullets before division is = 3 * X = 18

**Find sum of digits of D.**Let

A= 1999

^{1999}

B = sum of digits of A

C = sum of digits of B

D = sum of digits of C

(HINT : A = B = C = D (mod 9))

**Answer****The sum of the digits od D is 1.**

Let E = sum of digits of D.

It follows from the hint that A = E (mod 9)

Consider,

A = 1999^{1999 }< 2000^{2000 }= 2^{2000}* 1000^{2000 }= 1024^{200}* 10^{6000 }< 10^{800}* 10^{6000 }= 10^{6800 }i.e. A < 10^{6800 }i.e. B < 6800 * 9 = 61200 i.e. C < 5 * 9 = 45 i.e. D < 2 * 9 = 18 i.e. E <= 9 i.e. E is a single digit number. Also, 1999 = 1 (mod 9) so 1999^{1999}= 1 (mod 9)

Therefore we conclude that E=1.

**2. There is a 50m long army platoon marching ahead. The
last person in the platoon wants to give a letter to the first person leading
the platoon. So while the platoon is marching he runs ahead, reaches the first
person and hands over the letter to him and without stopping he runs and comes
back to his original position.**

In the mean time the whole platoon has moved ahead by 50m.

The question is how much distance did the last person cover in that time. Assuming that he ran the whole distance with uniform speed.

Answer

Answer

**The last person covered 120.71 meters.**

It is given that the platoon and the last person moved with uniform speed. Also, they both moved for the identical amount of time. Hence, the ratio of the distance they covered - while person moving forward and backword - are equal.

Let's assume that when the last person reached the first person, the platoon moved X meters forward.

Thus, while moving forward the last person moved (50+X) meters whereas the platoon moved X meters.

Similarly, while moving back the last person moved [50-(50-X)] X meters whereas the platoon moved (50-X) meters.

Now, as the ratios are equal,

(50+X)/X = X/(50-X)

(50+X)*(50-X) = X*X

Solving, X=35.355 meters

Thus, total distance covered by the last person

= (50+X) + X

= 2*X + 50

= 2*(35.355) + 50

= 120.71 meters

Note that at first glance, one might think that the total distance covered by the last person is 100 meters, as he ran the total lenght of the platoon (50 meters) twice. TRUE, but that's the relative distance covered by the last person i.e. assuming that the platoon is stationary.