This one I picked up from the net: A billion, in America, is a thousand million. That would be written as 1,000,000,000. If a book is published with only the dollar sign, printed 1,000 per page, it will be 1,000,000 pages long, and 2,000,000 inches thick!

There are 12 inches in a foot, and 5,280 feet in a mile, so dividing 2,000,000 by 12 to convert the inches to feet, and then dividing that by 5,280 to convert the feet to miles, our book will be166 feet 8 inches thick! Interestingly, in the UK, the British define a billion as a million million. That will be 1,000,000 times 1,000,000, which will be written as 1,000,000,000,000. How thick will our book be, if published in the UK? Approximately 166,667 feet thick. So our book of dollar signs will be over 31 miles thick!

PUZZLE 1: Find the smallest square with the last digit not 0, which becomes another square by the deletion of its last two digits. Find all squares, not containing the digits 0 or 5, such that if the second digit is deleted, the resulting number divides the original one.

PUZZLE 2: There are several settlements around Big Lake. Some pairs of settlements are directly connected by a regular shipping service. For all, A towards B, settlement A is directly connected to X if or exactly when B is not directly connected to Y, where B is the next settlement to A counterclockwise and Y is the next settlement to X counterclockwise. How many trips at most are required to move between any two settlements?

PUZZLE 3: A spider spins its web in a window frame. Each day it spins an area equal to that of the amount already completed. It takes 30 days to cover the entire window frame. How long would two spiders take? (In the case of two spiders, each of them spins an amount equal to the area of the web made by that particular spider).

Solutions on September 5

CORRECT ENTRIES

August 8

Abhisekh Ghosh, Malda; Subhash Kumar Chattopadhyay, Siliguri; Amitabha Roy Sharma, Bokaro Steel City; P.S. Sinha Roy, Dhanbad; P.S. Ananthakumar, Jamshedpur; A.R. Ghosh, Cal-91; P.K. Sen, Dum Dum Park; Arun Jyoti Talukdar, Sodepur.

CORRECT ENTRIES

August 1

Suhasini Chatterjee, Cal-58; R.K. Srivastava, Ranchi; Uday Kumar Acharyya, Alipurduar; A.K. Sengupta, New Jalpaiguri; C.K. Moorthy, Howrah; Abhishek Dasgupta, Bhagalpur.

Please send your entries addressed to knowhow@abpmail.com within 10 days. Postal address is Brainstorming, The Telgraph, 6, Prafulla Sarkar Street, Calcutta- 700 001. Don't forget to mention the date of the puzzles.Send complete solutions, not one line answers.

PUZZLE CRACKED

The puzzles published on August 8 didn’t seem to be complex at the outset. Judging by the number of responses, it appears it was not that easy.

Solution 1: m+-1 numbers.

Explanation: The m+-1 numbers in the first row and first column can all be chosen arbitrarily, but are sufficient to determine all the numbers. Hence at least m+-1 numbers must be left for the table to be accurately restored.

Solution 2: He must ask at least 11 questions to be sure of determining all the numbers.

Explanation: We can write the condition as b-a=d-c, so the 10 numbers in the first row and one in the second can all be chosen arbitrarily. Hence at least 11 questions are needed. But they are also sufficient. Having determined those numbers, the others immediately follow.