Alok Goyal's Puzzles Page

Many people sent correct answers, which I had expected. However, very few sent the explanation behind the answers, which is what I was looking for. Some of you used equations to solve the question. Intent was to solve the questions with no equations, and only use logical reasoning.

Ones to send the correct answer included Ajay Pandit (Bangalore), Anshi Aggarwal (Gurgaon), Radhika Goyal (New York), Karan Sharma (Jaipur), Pravin Panchagnula and Rhea Mittal (Mumbai).

Here is a way to think about both the questions without equations:

Answer #1

(i) The number is greater than 9 since there are two digits

(ii) It is less than 23 because 23 x 4 1/2 is greater than 100

(iii) It is an even number because it is an integer when multiplied by 4 1/2

(iv) Since the number is even, its half is an integer. When multiplied by 9, it yields the reverse and hence the reverse is divisible by 9

(v) Since the number has the same digits as the reverse, it implies that the number itself is divisible by 9

(vi) between 9 and 23, only number divisible by 9 is 18 which is the answer

Answer #2:

(i) None of the four integers is a 10 or a multiple of 5, if it was the product would end in a 0

(ii) At least one of the digits is less than 10, or the product would be 5 digits

(iii) All four are less than 10 [because of reason #(i)]

(iv) Therefore the only possibilities are 1,2,3,4 or 6,7,8,9. Since 1,2,3,4 yields only 24, 6,7,8,9 is the answer!

Hope you enjoyed the puzzle!

I got many correct answers for this puzzle. The first one to send the correct answer was Sandeep Verma from California. I also got correct answers from my niece Radhika, and from Girish Tutakne and Anubhav Garg.

I am reproducing the answer from Sandeep Verma, which is only slightly different than the one I am posting on video (BTW, LSD means Least Significant Digit and MSD means Most Significant Digit):

EFGH

X 4

——–

HGFE

Since multiplication of 4 and E in the MSD doesn’t lead to  carry over, E has to be either 1 or 2

Since LSD H*4 is E, E can not be 1. So, E=2

This implies H is either 3, or 8, since H*4 ends in E=2 being the units place. From the MSD, H of the result we know that H>4, so H=8

Now from middle two digits,

(G*4+3) mod 10 = F

LHS has to be odd, hence F is odd.

4*F+Carryover from above=G. Since there is no carryover to the MSD, E (4*2=8),

4*F+CO <= 9. Only values of 1 and 2 can satisfy this inequality for F.

As F has to be odd, F=1

Going back, G*4 + 3 = 11 V 21 V 31. G*4 ends in 8 in the units place. This means G=2 or 7. G*4+3 will be 11 or 31 respectively

For G=2, 4*(F=1)+1=2 is incorrect

For G=7, 4*(F=1)+3=7 is correct; hence G=7

Putting together, EFGH=2178 and HGFE=8712.

Moment of truth, 2178*4=8712

Here is a link to a video that explains the answer as well, which might be easier for children:

Hope you enjoyed the puzzle!