If you’re driving and notice that there’s a car in front of you with licence plates that read IXMNIZ. Can you guess the person’s occupation? If not, then:The current system of vehicle registrations in some countries is based on a registration number consisting of a letter for the year, followed by three digits, followed by a three-letter combination. In fact, some three-letter combinations are never used because they form offensive or vulgar words. If all three-letter combinations could be used, what would be the maximum number of vehicles that could be registered in one year, using this system? Also, by what number is the theoretical maximum reduced for each offensive three-letter combination that doesn’t get used?
THROUGHPUT
(The older but not much wiser problem was: “If you cut a rubber band to make one long strip and pull it from both ends in opposite directions will there always be a point on the strip that doesn’t move?”)
Let the rubber band itself constitute the x-axis. Then let f(x) be the distance point x have moved from its original position when the rubber band is stretched. Now, the left endpoint a will move to the left and the right endpoint b to the right so that f(a) < 0 and f(b) > 0.
As f is continuous and f(a) < 0 < f(b), there is at least one number x in the closed interval such that f(x) = 0. -- Saifuddin S F Khomosi, Dubai.
In a rubber band, the side by side and woven polymer strands are twisted and bent. When it is pulled from both ends, the strands on either side of a particular point get unbent and untwisted in opposite directions. The stretch on either side of that point is directly proportional to the force of the pull on the corresponding side. Greater the pull from one end longer the stretch from the said point to that end. That particular point doesn’t move. -- Abhay Prakash, abhayprakash@hotmail.com
(The second one was: “A man had to pack apples into packets but, as each packet had to have the same number of apples, he was having difficulty. If he packed 10 apples per packet, one packet had only 9. If he packed 9 apples per packet, one had only 8. If etc etc and so on down to 2 apples per packet. How many apples did he start with?”)
When we divide the total apples by 10, we get a remainder of 9 and so on. The number has to be 1 less than a multiple of 10; 1 less than a multiple of 9; 1 less than a multiple of 8, etc. So, let’s find the LCM of numbers 2 to 10 and subtract 1. LCM is 5*7*8*9 (other numbers are accounted within this) = 2520. This gives 2519 apples as the minimum number to start with. You can keep adding 2520 to this to get more such numbers. -- Saishankar Swaminathan, saishankar482@gmail.com
(Among the first five who also got it right are: Ilango Balakrishnan, ilangobalakrishnan@yahoo.com; Hemalatha T, hemalatha1956@gmail.com; Purushothaman Chandrathil, purushothamanchandrathil1969@gmail.com; Nrusingha Behera, ncb123.age@gmail.com; Rekha G, g.rekhapai@gmail.com.)
(The third problem was: “A tabletop two-pan scale exactly balances a glass of water against a certain weight. Would the balance get upset if you put your finger in the water without touching the glass?”)
When you dip your finger in the glass of water the balance will be upset because of two reasons: (1) Dipping displaces water of volume equal to volume of finger and creates an upward buoyancy which has a downward reaction on glass pushing it down; (2) For dipping finger you have to exert some pressure to overcome surface tension meaning you are exerting some downward force on glass of water -- Raghavendra Rao Hebbani, rao.raghavendrah@gmail.com
When a finger is dipped in the water, the water exerts upward thrust on the finger (Archimedes’s Principle). Due to Newton’s third law of motion, the finger will exert an equal and opposite force on the water in the downward direction resulting in an increase of the weight of water. Thus the balance will be upset & the glass of water will now need greater weight on the other side to balance. --Ajit Athle, ajitathle@gmail.com
BUT GOOGLE THIS NOW
1. What’s the next letter in this sequence: W T N L I T _?
2. How many colours are necessary to paint the squares of a chessboard so that no bishop can move between two squares of the same colour?
Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)