A casino manager wished to test the accuracy of his roulette wheel so he asked six croupiers to record the number of times each number came up in one evening to see if there was a bias on the wheel that favoured one particular number. He asked each of them to report to him which number came up the most times. The wheel was a standard one with 36 numbers, one through 36.
These were the answers that he received. (But note that two croupiers lied.) A: The number was even; B: The number was prime; C: The number was square or cube; D: The number was a double digit; E: The number had at least one ‘2’ in it; F: The number was between five and fifteen. So what was the biased number of the wheel?
THROUGHPUT
(The original problem was: “A is standing 100 metres south of B. B starts travelling east at speed x, whereas A starts travelling at speed 2x in a direction that always faces B. How much distance would B have travelled when he meets A?” The change was: what if their speeds were equal? How close could A and B get?)
If A also travels due east the distance will always be 100 meters as the two lines are parallel. If A travels due north for 100 metres and then turns towards east, then too the distance will be constantly 100 metres. If A travels towards north east from his original point, the distance becomes less and less up to a certain point. Let a and b the starting points of A and B respectively. Let c be the point where B will be at a particular point of time. A travelling towards north east at 45 degrees will meet the line bc and let that point be d. Now, abd will be an isosceles right angled triangle where ab = bd = 100 metres. The side ad will work out to be 141.421 metres. -- Dr P Gnanaseharan, gnanam.chithrabanu@gmail.com
(The second one was: “If you’re familiar with Christmas tree decorations you’re familiar with all those shiny little polished balls you hang on them, right? But how do you think one of those would reflect a point source of light in an otherwise darkened room?”)
The spherical reflecting ball is, basically, a convex mirror. And optics fundamentals teach us that any object placed in front of a convex mirror (irrespective of the distance from the mirror) forms an erect, diminished, virtual image behind the mirror, you could say between the principal focus and the mirror. That being so, when we have a point source of light as object, we will get a virtual point image behind the mirror. Thus, in a darkened room with a point source of light, it will appear as if there is a point source of light inside the reflecting ball. -- Balagopalan Nair K, balagopalannair@gmail.com
(The third problem was: “A cat and dog run a race, 100 metres straight and return. The dog leaps three metres at each bound and the cat only two metres, but then she makes three leaps to his two. Under these circumstances, what is the second answer to the question: Who wins the race?”)
The cat's each bound takes her 2m. She must make precisely 100 bounds to complete the race. The dog is at 99m at the end of 33rd bound. It must make one more -- ie, 34th bound to go past the 100m mark by 2m. Again it must make 34 bounds to finish the race -- ie, a total of 68 bonds. Now, the dog’s speed is 2/3rd of cat’s. When the cat is making its 100th jump, the dog cannot quite complete 67th. So, the hands-down winner is the cat. -- Saishankar Swaminathan, saishankar482@gmail.com
If the “she” which makes three leaps to his two, refers to the dog, then the dog actually covers 9 meters while the cat covers only 4 meters. In this case, the dog finishes the race in 68 leaps, whereas the cat would have completed 68*2/3 leaps and would have covered a distance of 90.6667 meters only out of 200 meters. Thus the dog wins the race by a huge margin of more than 109 meters. -- Narayana Murty Karri, k_n_murty@yahoo.com
BUT GOOGLE THIS NOW
1. 40,00,000/80,00,000 are the largest numbers that have this specialty. What is it? (Submitted by Sheikh Sintha Mathar, sheikhsm7@gmail.com) (Hint: Hardly nything to do with math – MS)
2. What’s a 15-letter English word containing all the vowels in which no letter is used more than once; and two eight-letter EWs that contain the first six letters of the alphabet.
Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)