To Coin A Face

                                      ....... and one for the gaol! 

Since we’re going to be on coins in a moment here’s something you can explain first. Put three coins in any order (except not the same face up for all). Did you know that flipping the left coin over once, then the middle one and finally the left again will always end with all coins with the same face up?
Okay so this is the game you’re being forced to play in prison. You get to toss a coin until it lands tails. The number of lashes you get equals the number of coins you end up tossing. (Meaning if you get a tails on the first toss, you get one lash; if you get one heads before a tails, you get two lashes, and so on and so on, etc.) How many lashes do you think you can actually expect to get?

THROUGHPUT
(The unsettled problem was: “A die with the numbers 1 to 6 is repeatedly thrown until the running total first exceeds 12. What’s the most likely total that will be obtained?” Yes you guys cheated shamelessly from Google but at least in future rephrase the answer intelligently instead of hashing up a cut and paste job.)

Consider the throw before the last one. Its running total must be 12, 11, 10, 9, 8 or 7. If it is 12, then the final total will be 13, 14, 15, 16, 17 or 18 with an equal chance for each. Similarly if the next to last total is 11, the final total is 12, 13, 14, 15, 16 or 17 with an equal chance for each and so on. Thus the number 13 appears as an equal chance candidate in all six cases. Thus the most likely total that will be 13.-- Narayana Murty Karri, k_n_murty@yahoo.com

The first instinct is to say 13. But the catch is the required total exceeding 12 must be most likely at the earliest. If a different number appears each time, the total is 21 in 6 throws. The expectation of total on each throw is 21/6 = 3.5. Total n requires a*n/3.5 = t throws where a is 1 if n/3.5 is an integer, otherwise a is 7 to make t an integer. Earliest n exceeding 12 becomes 14 with a = 1 and t = 4 throws. Other totals will require more throws. Number of throws required for 21 is 1*21/3.5 = 6. But, throws required for 13 is 7*13/3.5 = 26. Our answer is 14 which is expected in four throws. -- Abhay Prakash, abhayprakash@hotmail.com

(The second one was: “LEFT, OFF, SCREEN, DUST, SEED, OUT, TRIM. What’s common to all of them?”)
All the words are Janus faced words (also called contranyms or auto-antonym) in that they have two meanings opposite to each other. To give examples: (1) After many people left the party, only a few were left; (2) The alarm went off because it was not switched off; (3) The drapes that were not opened screened the wall; when the movie was screened; (4) Dust the mirror but dust the sugar on a cake; (5) Seeded dates have their seeds removed but seeded fields have seeds sown; (6) When the city lights were out, the stars in the sky were out; (7) Trimmed beard gives him a trim look. --  Dr Ramakrishna Easwaran, drrke12@gmail.com

(The third problem was: “If you placed a 6×6 cm square on a triangle, you could cover up to 60% of the triangle. If you place the triangle on the square, you can cover up to 2/3 of the square. What is the area, in square cms, of the triangle?”)The key is to see that these measures are one and the same: The triangle overlaps the square by the same amount that the square “underlaps” the triangle, so these areas are maximized at the same moment. If A is the area of the triangle, then 0.6A = 2/3 × 36, and A = 40 cm^2. -- Aruna Menon, mansuk78@gmail.com

When the square covers as much of the triangle, the triangle will also do the same. Hence, the amount of triangle covered is the same as the amount of square covered. Let T be the area of the triangle. Then 0.6T = 2/3*36 = T = 40 sq.cm. -- Saishankar Swaminathan, saishankar482@gmail.com

BUT GOOGLE THIS NOW
1. The two numbers 220 and 284 have got a special connection between them. What is it? (Submitted by Sheikh Sintha Mathar, sheikhsm7@gmail.com)2. Everyone has seen photgraphs of Mount Fujiyama in Japan reflected perfectly in the Kawaguchi Mirror Lake. In such images is there any way to make out the difference between the volcano and its reflection?

Sharma is a scriptwriter and former editor of Science Today magazine.(mukul.mindsport@gmail.com)

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