...... and a mad scramble!
(1) In this list there is a true statement and a false statement whose numbers add up to give the number of a false statement. (2) 2. Either statement #4 is false or there are three consecutive true statements. (3) The number of the last false statement subtracted from the product of the numbers of the first and the last true statements gives the number of a statement which is false. (4) The number of even-numbered true statements equals the number of false statements. (5) One of the first and last statements is true and the other is false. (6) When I first sent this problem in, thanks to a typing error, no sixth statement was included. However, the answer to the following question was same then as it is now: What statements are false?
(PS: Too tough? Go to both BGTNs at the end and a have a good time thinking how easy things can get in life.)
THROUGHPUT
(The older slightly misunderstood problem was: “A triangle whose sides are consecutive integral lengths has all its angles acute. What could be the minimum integral area of such a triangle?”)
A triangle with three consecutive integers as its sides and whose area is also an integer is called a Super Heronian triangle. Minimum integral area of such a triangle, with all three angles less than 90 degrees, and with sides as 13,14 and 15 is 84 sq.units. -- Seshagiri Row Karry, srkarry@yahoo.com
The smallest triangle, whose sides are consecutive integral lengths with all its angles acute and with the minimum integral area, has sides 13, 14 and 15 units. The area of this triangle is 84 square units. Substituting natural numbers for n, the first (smallest) value that satisfies this equation is 2, which gives x = 4. The triangle, then, has sides of lengths 3, 4 and 5 units. This cannot be the solution, because it is a right angled triangle. -- Balagopalan Nair K, balagopalannair@gmail.com
(The second one was: “Charles takes part in a 14 stage cycle race in which there are 100 competitors. In all 14 stages Charles finishes in the 93rd position. Yet he’s declared the runner up. Etc, etc, etc.”)
Other than Charles and the winner there are 98 other participants (100 - 2). If all the 14 races had seven persons after Charles (positions 94 to 100) and these were all different across the 14 races (14*7 = 98), that would mean they had times worse than Charles in that race. If their time in that one race was much larger than Charles, then across the 14 races, Charles could have a better time than all of them. -- Kishore Rao, kishoremrao@hotmail.com
The answer lies in the rules of multi-stage cycle racing. A minimum time is assigned for each stage for the cyclists. Anyone who fails to achieve the minimum time of any stage gets disqualified. This must have happened with the 98 cyclists other than the runner-up and the winner. The one who was in 93rd position must have achieved the minimum time in all laps or stages of the race. -- U N Murthy, u_n_murthy@rediffmail.com
Charles being 93rd in each race lead seven racers in each race. If the seven that trailed were different in each race (7*14 = 98) leaving ne ahead always and if he beat all the 98 once but with bigger time margin than any one’s total lead time over him in rest of 13 races, he could’ve been runner-up. -- Raghavendra Rao Hebbani, rao.raghavendrah@gmail.com
(Among the other five who also got it correct in good timing are: (K K N Raj, kknraj2017@gmail.com; Dr P Gnanaseharan, gnanam.chithrabanu@gmail.com; Narayana Murty Karri, k_n_murty@yahoo.com; J Vaseekhar Manuel, orcontactme@gmail.com; Abhay Prakash, abhayprakash@hotmail.com.)
(The third one wasn’t there this time. Sorry about that!)
BUT GOOGLE THIS NOW
1. We all know how to get the Error symbol ‘E’ on a calculator. Now the question is, how will you get two such symbols consecutively as ‘EE’? – (Submitted by Sheikh Sintha Mathar, sheikhsm7@gmail.com) 2. A number ends with the digit 2. If we move this 2 from the last place to the first, the new number is twice the original. What’s the number?
(PS: Both too easy? Go to the beginning and have a wretched bad hair day with a broken comb in your hand.)