Saturday, June 1, 2002
M I N D  G A M E S


Make yourself a super sandwich

Students always have to be taught what they should have learned in the preceding course. The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra, he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it; the most advanced course is learned only by teaching it. It is because there is no time for enough practice on each new topic. — Ralph P. Boas

The CBSE Chairman, Ashok Ganguly says that, from this session, mathematics will be taught in a fun way in schools, with laboratories for experimentation. Way to go chief.

SATURDAY mornings set the mood for the rest of the weekend, so, these should ideally be like perfectly baked cakes - with just the right amount of warmth and icing. Hmmmmm… I can smell taste buds getting tickled, so, let us quickly find something to eat. The refrigerator is empty, but don't lose your cool because we are going to make ourselves a hot new dish. Let's see, what is it that everyone eats with delight. A sandwich is always a good idea for breakfast - one can eat it as one likes.

Today's recipe is Groovy Sandwich. To make it, we need all digits from 1 to 9, a little bit of mind and some concentration. Now, we add some spice to it. 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. If you are a person with elephant appetite, we have for you Jumbo Sandwich - a number where the sandwich condition is met, but each digit occurs three times or more, rather than twice.

Taste a single-decker 8-sandwich to get the ingredients right — 151847652432873.

 


It has one digit (5) between two 1s to begin with, followed by eight digits (47652432) sandwiched between two 8s, four digits sandwiched between two 4s, 7 digits sandwiched between two 7s, three digits sandwiched between two 3s and two digits sandwiched between two 2s. There are 150 such 8-sandwiches, besides their mirror images. Making triple-deck-8-sandwiches will perhaps take empiricists a long, long time in the kitchen, but I would appreciate if someone could make it for me and give me the recipe as well.

While making triple-deck-sandwiches, the aspiring chefs should remember that for some values of n, there are triple-deck-n-sandwiches; and for other values of n, there are none. For which values of n do triple-deck-n-sandwiches exist and for which values do they not …and why?

The number 312132 and its mirror image 231213 are perhaps the only 3-sandwiches, or are we wrong? You cannot make the same with only ones and twos, because between the two twos there must be two digits, which have to be ones as these are the only available digits, but that means that between the two ones there are no digits, which is not possible. The only 4-sandwiches are 41312432 and its mirror image. You can make a lot of 7-sandwiches because there are 26 of these, besides their mirror images. There are no n-sandwiches for some values of n; which ones? Send in your recipes at The Tribune or adityarishi99@yahoo.co.in.

— Aditya Rishi