The ruins of a tower which once served as a lighthouse stand upon a little ledge of rocks that runs out into the sea. Records show that it was originally 300 feet high. The centre support of the structure was composed of huge poles skilfully spiked together, about which there wound a spiral staircase with an iron handrail. The rail went exactly four times around the column. There was one baluster (supporting post of the handrail) to each step, and as these were just one foot apart, it’s a simple matter to find out how many steps one had to take to reach the top.
To summarise: The tower was 300 feet high from the ground to the top of the last step, the handrail circled the tower four times and the balusters in the rail -- one to each step -- were one foot apart. Add to this that the diameter of the entire tower (ie, the dia of the imaginary cylinder around which the rail twines) was 23 feet, 10.5 inches. How many steps were on the circular staircase?
THROUGHPUT
(The non-leftover problem was: “Why does the light in a kerosene lantern become erratic when the glass globe is removed?”)
In a typical lantern the glass globe sits on a shallow metallic cup with holes around its edges. This ensures the flame gives a steady output of light due to a uniform supply of oxygen which is maintained by convection currents set up in the globe by cold air drawn through the holes at the bottom and leaving through the opening at the top. In the absence of the globe the flame gets disturbed by drafts and an uneven supply of oxygen. -- Mridula Gupta, mdgupta431@gmail.com
In a kerosene lamp the fresh and cooler air (oxygen) enters from holes at the bottom to help combustion of kerosene. The burnt gases escape from the open top of the glass chimney. When this glass cover is removed the flame is disturbed by the air blowing across though getting oxygen supply. Hence the wavering of the flame. -- U N Murthy, u_n_murthy@rediffmail.com
When the glass globe is removed, air flows in from all directions uncontrollably thereby making the flame flickering erratically. -- Shashi Shekher Thakur, shashishekher@yahoo.com
(The second one was: “A digital clock shows both date and time in the format hh:mm:ss month/day. It is a 24-hour clock and times range from 00:00:00 to 23:59:59. If all 10 digits 0 to 9 appear on the display, what is the earliest and latest date and time in the year this could happen?”)
The solution to the problem using 0 to 9 is: earliest time: 17:48:59 03/26; latest time: 17:56:43 09/28. We can arrive at the solution by putting constraints like HH should be less than 24, MM and SS less than 60, month less than 13, date less than 31, first H should be either 0, 1 or 2, the first digit in month should be 0 or 1. -- Ravi Nidugondi, ravi.nidugondi@gmail.com
A really tantalising problem, warranting many trial and error attempts. However, some limitations such as the first digit of the hour can be only a maximum of 2 and that of the month a maximum of 1, help to a great extent. 17:48:59 of 03 (March)/26 is the earliest and 23:58:47 of 09 (Sep)/16 is the latest duration in the year involving all the digits 0 thru 9. But still not confident about the correctness of my solution. -- Sheikh Sintha Mathar sheikhsm7@gmail.com
(The third problem was: “A triangle whose sides are consecutive integral lengths has all its angles acute (no angle 90 degrees or more). What could be the minimum integral area of such a triangle?”)
The triangle in question is a Heronian triangle (with integer sides and integer area). One of the properties of a Heronian triangle is that its area is always divisible by 6. Moreover, there are no Heronian triangles with a side length of either 1 or 2. Thus, the minimum integral area of such a triangle has to be 6 (with sides 3, 4 and 5). -- Saifuddin S F Khomosi, saif_sfk@hotmail.com
BUT GOOGLE THIS NOW
1. Why do decomposing animals such as birds or dogs smell so bad whereas dead insects or ants do not? (More importantly, what’s the second reason?)
2. What’s the bet that, like me, you also vaguely didn’t know that snake venom is not used directly to treat a snake bite. Instead . . . what?
Mukul Sharma
Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)