### By Mukul Sharma

Published: Fri 25 Dec 2015, 12:52 PM

E4
Archimedes (of the water screw fame) said, "Compute, O friend, the number of the cattle of the sun which once grazed upon the plains of Sicily, divided according to colour into four herds, one milk-white, one black, one dappled and one yellow. The number of bulls is greater than the number of cows, and the relations between them are as follows: white bulls = (1/2 + 1/3) black bulls + yellow bulls; black bulls = (1/4 +1/5) dappled bulls + yellow bulls; dappled bulls = (1/6 + 1/7) white bulls + yellow bulls; white cows = (1/3 + 1/4) black herd; black cows = (1/4 + 1/5) dappled herd; dappled cows = (1/5 + 1/6) yellow herd; yellow cows = (1/6 + 1/7) white herd." Then as if that was not enough, Archie goes on to say, "white bulls + black bulls = a square number; dappled bulls + yellow bulls = a triangular number." (Friendly advice: Try not to attempt it because the answer turns out to be more cattle than the entire universe can hold! Try the following down-to-Earth one instead.)
Consider the following two-person game, in which players take turns colouring the edges of a cube. Only three colours - red, green and yellow - are available. The cube starts off with all the edges uncoloured and, once an edge is coloured, it cannot be coloured again. Two edges with a common vertex are not allowed to have the same colour. The last player to be able to colour an edge wins the game. Given best play on both sides, should the first or second player win? And, importantly, what is the winning strategy? (Meaning, no chancing the answer.)
DEAR MS
(The problem from two weeks ago was: "Of those numbers whose English representation in capital letters consists only of straight lines, only one number has a value equal to the number of straight line segments required to write it out. What number?" - MS)
29-Times-Lines Dept:
The number is TWENTY-NINE. - Joel John, joeljo2104@gmail.com
The number whose value is equal to the number of straight lines in its English representation is TWENTY-NINE. - Tasneem Murtaza, tasneem.murtaza.s@gmail.com
The number is TWENTY-NINE. However, the only other number to possess this property, if its representation is permitted to have even curved letters, is SIXTEEN. - Sheikh Sintha Mathar, sheikhsm7@gmail.com
(The second problem was: "Given are 13 five-lettered nouns, each of which has had two of its letters removed. In total, they comprise 26 letters from A to Z. The remaining letters in each word are in correct order. What are the original words? CLH, DIR, INE, EBA, UAL, TOE, YAT, COH, REA, OKR, AOR, REE, SAM." - MS)
Letters-To-The Editor Dept:
The original words are: CLOTH, DIARY, INDEX, ZEBRA, QUAIL, STOVE, YACHT, COUGH, FREAK, JOKER, MANOR, REBEL and SWAMP. - Saifuddin S F Khomosi, saif_sfk@hotmail.com
The original words are: CLOTH (OT); DIARY (AY); INDEX (DX); ZEBRA (ZR); QUAIL (QI); STOVE (SV); YACHT (CH); COUGH (UG); FREAK (FK); JOKER (JE); MANOR (MN); REBEL (BL); SWAMP (WP). - Surya Narayanan Krishna Moorthy, surya661666@yahoo.com
(The third problem was: "How many numbers between 1 and 1,000,000 have the sum of their digits as 18?" - MS)
Going-On-Eighteen Dept:
25,927 numbers between 1 and 1,000,000 have the sum of their digits as 18. One method of solving this is by using combinations and the other is by designing a simple program using C++, compiling it and then running it. - Shashi Shekher Thakur, shashishekher@yahoo.com
ENDGAME(S)

1. What are the last two numbers in this sequence of seven numbers: 103, 504, 10, 62, 1013, .  (Hint: It's entirely logical and related to Romans.)
2. A farmer plants a tree in a square plot of land so that it is 100m and 140m away from the two opposite corners of the plot and 60m away from the third corner. What is the area of the plot?
(To get in touch with Mukul, mail him at mukul.mindsport@gmail.com)

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