THERE is a reason to cheer for fans of mathematics — ‘A Beautiful Mind’ by Ron Howard featuring Russell Crowe has won the Oscar for the Best Film this year. It is a story of a schizophrenic mathematician John Nash who is also a Nobel laureate. Some time ago, in this column, I had written that more mathematicians and chess players go mad than artists and poets; this year’s Best Film at the Oscars proves it.

A lot among you have sent in correct solution to the Bond problem and the race to the Oscar for the Best Solution was close and the competition huge. Most of you with computers, perhaps, did a quick hit and trial to arrive at the answer — poor Bond, he had not time for this; he had to be accurate the first time, with no computer to help him. A computer only gives you the digit, but does not tell you how it has cracked the problem.

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Most correct solutions identified that groups of six 8s and six 9s that are divisible by seven. (All six-figure numbers made by repeating the same digit are divisible by 7). Because the original number had 50 digits on each side of M, there are eight groups of six and two digits left, which reduces the problem to: What is M if 88M99 is divisible by 7?

When 8,88,888 is divided by 7, we get 1,26,984. Besides, 8, 88, 888, 8888 and 88888 cannot be divided by 7 without any remainder and 8,88,888 is the smallest number made up of eights that can be divided by 7. When the digit 8 is repeated 50 times and divided by 7, we will get the number 1,26,984 repeated 8 times and 12 behind the number, which is 12698412698412698412698412698412698412698412698412 and has remainder 4.

When 9,99,999 is divided by 7, we will get 1,42,857. Besides, 9, 99, 999, 9999 and 99999 cannot be divided 7 without any remainder and 9,99,999 is the smallest number made up of nines which can be divided by 7. When the digit 9 is repeated 50 times and divided by 7, we will get the number 1,42,857 repeated 8 times and 14 behind the number, which is 142857142857142857142857142857142 85714285714285714 and has remainder 1.

When digit 8 is repeated 50 times and divided by 7, the remainder 4 is carried forward and, in order to avoid getting any remainder when the digit 9 is repeated 50 times and divided by 7, we have to take out the first two 9s. Therefore, 4M99 must be able to be divisible by 7, so after such trial and error, most of you find that the number that can be divided by 7, is 4599, which give the value of the digit M to be 5. This is the smartest approach among all solvers that have written to me. If you were Bond and had a paper and pencil with you, you could have found out the digit, but is there a way to eliminate trial and error fully? Dr Lokesh Handa says that this 101-digit number is interesting; if we substitute every third digit, except M, in this number with any digit from 0 to 9, the value of M will remain same; and if the number is divisible by 11 or 13, the value of M would be 0 and 6, respectively.

Nevertheless, the Oscar goes to Tapasya, Dr Lokesh Handa, Kulpreet Singh, Nitin Somani, Saurabh Aggarwal of Batala, Amit Gangwani, Master Amandeep Jindal of Ludhiana, Nirmal, Ravinder Mittal of Ludhiana, Sukhrenjit of Amritsar, Pankaj Singla of Kaithal, Shubhangi Arora of Kurukshetra, Tajinder Singh of Ludhiana, Aman Preet, Puneet Sharma of Dasuya, Pawan Singhania, Lokesh of Patiala, Navneet Aggarwal and Suhaildeep Singh.