Saturday, August 16, 2003
M I N D  G A M E S

Transylvanian rescue
Aditya Rishi

God exists since mathematics is consistent, and the Devil exists since we cannot prove it.

Andre Weil (1906-1998)

HAKER asked Dracula and now Dracula wants the smallest whole number which, when multiplied by 7, gives a product made up entirely of ones. In rains this year, we led a rescue mission to Transylvania and went over the old ground which was to us so full of vivid and terrible memories. It was almost impossible to believe that the things we had seen and heard were living truths. Every trace of all that had been was blotted out. The castle stood as before, reared high above a waste of desolation.

Known through legend in such diverse countries as Greece, Rome, France, India and even China, he doth fatten upon the blood of the living. He lives by night, for by day he must conceal himself within his grave. Those upon whom he nourishes himself sicken and die and, then, become like him, vampires themselves. His name is Dracula, and now he has run away.


He is no longer feared, all due to Harker's brave rescuers most of whom defined a Repeating Unit as a number made up entirely of 1s so that R1 = 1; R2 = 11; R3 = 111; R4 = 1111 etc. Investigating by division: 11/7 = 1.571...; 111/7 = 15.857...; 1111/7 = 158.714...; 11111/ 7 = 11587.28...; 111111/7 = 15873. So, 15873 is the smallest whole number which, when multiplied by 7, gives a repeating unit. Extending the question to find what other numbers will also work, we know 3 must be the smallest number to work because a number ending in 1 will be odd and it will not divide by 2. 11/3 = 3.666...; 111/3 = 37. The number 4 will not work for the same reason 2 does not work, and 6 won't work, neither will 8. Any number multiplied by 5 will end in 0 or 5, so 5 won't work, so 9 will be the next number to try.

Amarjeet Kaur of Ludhiana started dividing by 7 taking dividend firstly 11. Remainder is 4. Then, she takes dividend 111, now the remainder is 6, takes 1111, now the remainder is 5, takes 11111, and the remainder is 2, takes 111111and the remainder is zero, and the answer is 15873. The second smallest whole number is 15873015873, which is 1000001 times larger than the first one, says Dr Vikas Handa. To have 1 in the product, you may even start by selecting digit 3 at one's place in the required number. We carry over 2. To have another 1, we should look for a product of 7 with 9 in one's place, which is 49; carrying on, we reach 15873.

The following rescuers were the first ones to arrive: Vikas Vashisht, Dr Manjit Singh, Parveen Bajaj, Pareena Bawa, Sourabh Kumar Kakkar, Navneet Kad (Ludhiana), Sunny Ahuja (Amritsar), Anil Chaudhry, Rashmi Anand, Rakesh Anand, Ajit Partap Singh (Ludhiana), Abhi Rana, Prerna Aggarwal (Panchkula), Nitin Khanna, Kamal Sethi (Hisar), Sanjay Chawla, Jaskirat Singh Jassal (Amritsar), Bhawna Garg (Mandi Gobindgarh), Mrinal Goyal (Kurukshetra), Divyadeep Modi (Malerkotla), Dr Tarsem Lal (Khanna) and Sameer Madan. There was so much mail that even Dracula got sacred and ran away. (Dedicated to Bram Stoker. Write at The Tribune or