THE signboard says: "Crocodile waters; do not enter", yet I have seen many men take the plunge. A lot of you turned away after reading the sign, but some brave men and women listened to their heart and put their head on the line. This service is in memory of such brave brothers and sisters, who could not cross the waters, but tried.

In the puzzle, the Harvard crocodile was in a race in which the starting line was at a certain point on a straight beach. The finish line was in the water. One way to arrive at the finish line is to run 4 km down the beach, make a 90-degree turn and swim 1 km. However, it may cut into the water at any point. Its speed on land is 6 kmph and its speed in water is 2 kmph. You had to tell at what point, measured from the starting line, should it cut into the water for an ideal run?

The correct, but still incomplete solution has been received from a person who has not identified himself or herself. The greatest heroes are the ones who do a great deed and slip away quietly…if you could tell me about you, Phantom, the world would have learnt of its hero. The answer is on a competition postcard, in a neat hand, but with no address at the end — a case for Sherlock Holmes, perhaps. This person has given the shortest, yet most effective solution, though I still insist that the knowledge of calculus is not required to solve this problem. This person (Phantom to all of us) writes: "Let y be the solution. Let x=4-y. The time to reach the finish line is: t=(4-x)/6 + (1+x2)1/2/2. Set the derivative equal to 0: dt/dx= -1/6 + 1/2 * 1/2 * 2x * (1+x2)-1/2 = 0. The values of x and y follow from here." This is true indeed, but Phantom, please take off your mask and reveal your true identity to us. Your solution certainly has the virtue of brevity.

Enough of this Phantom menace; we now move on to the others who have tried to solve the problem: Alka of Dashmesh Avenue, Jalandhar, has written a long, well-illustrated solution and used a bit of calculus as well to arrive at the figure x=3.6465 km. "The fastest way to reach at the point that the big croc mentioned is that the kid runs 3.646 km down the beach and, then, swims towards that point at an angle of 70.53 degrees (a distance of 1.061 km), when the time taken will be 62.29 minutes. This way, he can save full 1.71 minutes as compared to the time taken if he follows the path that the big croc suggested. (Maybe, the water flow could have helped him save more time, but the big croc forgot to mention the water flow rate (It is not required badly: AR)," says Rajeev Kumar Tak.

"The little croc should enter water after covering 3.2/3 km (Three and two thirds of a km) on land (one third of a km before the end," says Ravinder Mittal. Pavneet Singh finds the suggestion of the big croc the ideal way to finish the run." You can well imagine how close of far all of you have been to an ideal finish. Once again, it has taken a superhero to bail us out of troubled waters. Phantom, the immortal Ghost Who Walks, has vanished into thin air again, from where, he or she will surely return one day to take on another challenge. The Harvard reptile thanks you for saving his reputation. You remind me of a certain Mr Kit Walker, whose only contact was: c/o Postmaster, Morris Town, Denkali. Write at The Tribune or adityarishi99@yahoo.co.in.