Saturday, July 13, 2002 

HAMLET was sandwiched between to be or not to be; the Prince of Denmark was not to be a mathematician, but sandwiches can be a delightful breakfast, as proved earlier. You don't have to be a master chef to create numbers that have spice put inside; just look at the recipes that some readers have sent us. Sandwiches are made under
the conditions that, between any pair of ones in the list there is one
digit, between any pair of twos there are two digits, between any pair
of threes there are three digits and so on. An extension to the sandwich
number problem is to construct lists where the same sandwich condition
is met, but each digit occurs three times or more. Apart from the
unusual 000, there are 24digit sandwiches containing three each of the
digits 1 to 8. 
The readers have found a lot of 7sandwiches because there are 26 of these, besides the mirror images of these. They may have tried and failed to find 5sandwiches and 6sandwiches because there aren't any. There are no Nsandwiches for N=1, 2, 5, 6 or 9 or any number that leaves a remainder of 1 or 2 when divided by 4, a catch that no one took. Vineet Aggarwal, a software consultant, has this recipe: "I tried to make tripledeck8sandwiches, but found none. The tripledeck sandwiches are possible only with 9 and are: a) 181915 26728 52964 75384 639743; b) 191218 24627 94586 34753 968357; c) 191618 25726 92584 76354 938743; d) 347839 45367 48529 62752 816191; e) 347936 48357 46925 82762 519181; f) 753869 35743 68549 72642 812191; Also, 9sandwiches are not possible (except tripledeck ones)." He has made a small sandwich maker, a program in 'C' to do the trick. Now, he can generate anydecked sandwich. The program works. Dr Lokesh Handa, too, has a recipe: He gives us a tripledeck8sandwich: 784516 147583 642372 486131 753684 257246 181572 632583, which took him a gruelling anhourandahalf to make. "If a problem is approached scientifically, you get the results. I started with six 8's separated by 8 blanks. To put 7s in, there were two options: adjacent to first 8 or with a difference of one space... or even to directly start with 8. By trying out various combinations, the result can be seen. In the above number, if you put the first 7 at the end, you get a new sandwich," he says, "I have a doubledeck8sandwich: 645783 465317 182412 174682 53276358. The values of n, for which, there are no sandwiches, are doublenumbers like 11, 22 and so on, because these break the rule of placing a digit in between." They arrive at the top, but leave some trails unexplored. Write at The Tribune or adityarishi99@yahooo.co.in. — Aditya Rishi
