+0

# Hard math puzzles, only attempt if you dare!

+1
2
10
+38

I'll post the answers after 8 posts of attempts.

1. A 3 digit number is such that its tens digit is equal to the product of the other two digits which are prime. Also, the difference between its reverse and itself is 99. What is the sum of the three digits?

2. Stranded on a deserted island, Harry Puttar is left with only a 40 litres container of milk. To conserve his milk he decides that on the first day he will drink one litre of milk and then refill the container back up with water. On the 2nd day he will drink 2 litres and refill the container. On the 3rd day he will drink 3 litres and so on... By the time all the milk is gone, how much water has he drunk?

3. One side of the bottom layer of a triangular pyramid has 11 balls. How many are there in the whole pyramid? Note that the pyramid is equilateral and solid.

4. Substitute numbers for the letters so that the following mathematical expressions are correct.

ZYX/3 = LQ, PQR/6 = LQ, JKL/9 = LQ

Note that the same number must be used for the same letter whenever it appears.

5. When Poonia died, he willed his 17 dogs to the sons, to be divided as follows:

First Son to get 1/2 of the dogs, Second Son to get 1/3rd of the dogs & Third Son to get 1/9th of the dogs. The sons are sitting there trying to figure out how this can possibly be done, when a very old wise man goes riding by. They stop him and ask him to help them solve their problem. Without hesitation he divides the dogs properly and continues riding on his way. How did he do it?

Also, I am thinking about doing more of these posts, so every week, look out for problems!

Aug 10, 2024
edited by AUnVerifiedTaxPayer  Aug 10, 2024

#1
+1790
+1

1. Alright, let's give this a shot. First off, there are 4 prime numbers between 1 and 10.

We have $$2,3,5,7$$

However, since the product of these two prime numbers must be less than 10 (single digit number), the only two numbers that qualify is $$2 \cdot 3 = 6$$

So now, we have two different numbers to work with. $$263, 362$$

Since their the reverses, take the larger number, 362 is the original.

We find that $$362-263=99$$, meaning we found our number.

Adding the digits up, we get

$$2+6+3 = 11$$

Thanks! :)

Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
#2
+1790
+1

2. Note that if Harry Puttar (nice name LOL) refills 2 litres and drinks 3 litres, he will drink 1 litre of milk every single day.

Thus, he should run out of milk by the 40th day.

The number of litres he drank can be calculated with the sequence

$$1+2+3+4+5+6+...+39+40 = 820$$

Of these total 820 litres, there WAS 40 litres of milk,  meaning he drank

$$820-40 = 780$$litres of water.

Thanks! :)

Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
#3
+1790
+1

3. My logic for this problem might be flawed, but let me give it my best shot.

If One side of the bottom layer of a triangular pyramid has 11 balls,

The first layer has 1 ball

The Second layer will have $$3 = (1 + 2)$$balls

The Third layer will have $$6=(1 + 2 + 3)$$ balls.

The Fourth layer has$$10=(1 + 2 + 3 + 4)$$ balls.

The Fifth layer will have $$15=(1 + 2 + 3 + 4 + 5)$$ balls

The sixth layer there are$$21=(1 + 2 + 3 + 4 + 5 + 6)$$ balls

So there will be $$28,36,45,55 \quad \text{and} \quad 66$$ balls in the remaining layers.

So, adding the total number of balls up, we get

$$1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 = 286 balls.$$

Thanks! :)

Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
#4
+38
+1

I will give you that you got the three right! Nice job, and the method for number 3 was my strategy too. Great job NotThatSmart!

Aug 10, 2024
#6
+1790
+1

3. First, let's note that If any number LQ is multiplied by 9, right most digit of the product will be (10 - Q).

So, from JKL/9 = LQ, we get, L = 10 - Q

LQ can be mathematically represented as$$10L + Q = 10(10 - Q) + Q = 100 - 9Q$$

Since $$PQR/6 = LQ$$$$PQR = 6LQ = 6(100 - 9Q) = 600 - 54Q$$

We now try various values for Q such as 1, 2, 3, .... , etc, till the middle digit of the product turns out to be the same as the value of Q used. We get the correct answer for Q at the very third attempt, namely 3. Hence,

$$L= 10 - 3 = 7$$

So we finally have

$$219/3 = 73 \\ 438/6 = 73 \\ 657/9 = 73$$

Thanks! :)

Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
#7
+1790
+1

5. I THINK this is the solution. Unless,umm..idk

The old man temporarily added his dog to the 17, making a total of 18 dogs.

First son got 1/2 of 18 = $$1/2\cdot 18 = 9$$

Second son got 1/3 of 18 = $$1/3 \cdot 18=6$$

Third son got 1/9 of 18  = $$1/9 \cdot 18 = 2$$ for a total of 17.

Then I think the old man stole his dog back...so yay?

Nice questions! I might not answer them every week, but it sure was a nice little brain teaser!

Thanks! ::)

Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
#9
+38
+1

Alright, here are the solutions that I got.

1. It is given that the two digits of the required number are prime numbers i.e. 2, 3, 5 or 7. Note that 1 is neither prime nor composite. Also, the third digit is the multiplication of the first two digits. Thus, hundreds digit and units digit must be either 2 or 3 i.e. 2_2, 2_3, 3_2 or 3_3 which means that there are four possible numbers - 242, 263, 362 and 393.

Now, it is also given that - the difference between its reverse and it is 99. So 263 and 362 satisfy this condition. Hence, the sum of the three digits is 11 in each case.

2. It is given that the man has 40 litres container of milk. Also, he will drink 1 litre on the first day and refill the container with water, will drink 2 litres on the second day and refill the container, will drink 3 litres on the third day and refill the container, and so on till 40th day. Thus at the end of 40 days, he must have drunk (1 + 2 + 3 + 4 + ..... +38 + 39 + 40) = 820 litres of liquid.

Out of those 820 litres, 40 litres is the milk which he had initially. Hence, he must have drunk 780 litres of water.

3. As there are 11 balls along one side, it means that there are 11 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 and 66 balls in the remaining layers.

Hence, the total number of balls is = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 = 286 balls

4. Let's start with JKL = 9 * LQ. Note that L appear on both the side. Also, after multiplying LQ by 9 the answer should have L at the unit's place. The possible values of LQ are 19, 28, 37, 46, 55, 64, 73, 82 and 91; out of which only 64, 73 and 82 satisfies the condition. (As all alphabets should represent different digits)

Now, consider PQR = 6 * LQ. Out of three short-listed values, only 73 satisfies the equation.

Also, ZYX = 3 * LQ is satisfied by 73.

Hence, Z=2, Y=1, X=9, P=4, Q=3, R=8, J=6, K=5, L=7

219/3 = 438/6 = 657/9 = 73

5. The old man temporarily added his dog to the 17, making a total of 18 dogs.

First son got 1/2 of it = 9

Second son got 1/3 of it = 6

Third son got 1/9 of it = 2 for a total of 17.

He then steals his dog back and goes away......

Aug 10, 2024
#10
+1790
+1

Looks like we got the same answers! AUnverifiedTaxPayer!

Nice work! :)

Also, I accidentally labeled #4 as #3. My bad.

NotThatSmart  Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024
edited by NotThatSmart  Aug 10, 2024