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Problematics | Hidden treasure
Some text in a treasure map is illegible. How can you overcome this difficulty and find the hidden spot?
David Wells, a British puzzler and popular mathematics writer, may not be as well-known as Henry Ernest Dudeney, but he has played a significant role in popularising puzzles nevertheless. He has written books with mathematical musings and puzzles from sources around the world, including the Bhakshali manuscript from India. Among his writings I have found the following puzzle, whose source I am not sure about. The solution he provides, however, is attributed to the National Council of Teachers of Mathematics, USA (1965).
#Puzzle 186.1
Four towns are located along a straight road. If we call them A, B, C and D, then AB is 5 km, BC is 8 km, and CD is 11 km.
Now there is a treasure located at some point on the road. There is a map with instructions, but some of the writing has become illegible. In effect, what you can read is something like this:
“Start at town (illegible text #1) and travel half the distance to (illegible text #2). From there, travel one-third of the distance to (illegible text #3). Finally, travel one-fourth of the distance to (illegible text #4). The treasure will be at the end of this leg of the journey.“
To pre-empt any misunderstanding, all distances are relative to the point from which that leg of the journey is to be undertaken. For example, when you have completed the first leg from (illegible text #1), the second leg [“one-third of the distance to (illegible text #3)”] means you travel 1/3 of the distance between (your new position) and (illegible text #3). And so on.'
Despite the illegible text, as Wells shows, it is possible to locate the site of the treasure.
Where is the treasure? (Hint: begin the journey at some fixed point and see where the variables take you.)
#Puzzle 186.2
We often speak about running around in circles, metaphorically as well as mathematically. Now here is a somewhat unusual puzzle about running around a triangle.
Two runners, one inexperienced but promising and the other retired but fit, meet at one of the vertices of an equilateral triangle whose dimensions we are not told. The retired athlete’s running speed is 5 km/h and walking speed is 2 km/h. The younger one runs at 10 km/h and walks at 4 km/h.
At the vertex, the two agree that the older athlete will run while the younger one will walk. They start off from that point, the younger one going clockwise and the older one anticlockwise. After some time, both realise that the retired athlete has covered significantly more ground than the young athlete.
“Sir, can I run now while you walk?” says the younger one.
“Sure, young man, why not?” the other replies.
Without wasting a second, the younger athlete starts running while the older one starts walking, each continuing in his original direction. As it turns out, both come back to the starting point after exactly 45 minutes.
Where and when (a) did they change from walking to running (and vice versa) and (b) did they meet along the course?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 185.1
For possibly the first time for any puzzle in Problematics, the number of wrong answers has exceeded the number of right ones. In #185.1, it is possible to determine the ribbon colours of exactly two sisters out of four. While most readers have shown that Lulu’s ribbon is certainly red, only three have shown that Lily is certainly wearing blue.
Before going into the solution, let us recap the five given conditions so that we can refer to them by their assigned numbers:
(1) Lily has exactly two sisters wearing red ribbons.
(2) Leela has exactly two sisters who are not wearing a blue ribbon.
(3) Leena has exactly two sisters whose ribbons are either red or orange.
(4) At least one of Lulu’s sisters is wearing an orange ribbon.
(5) At least one of Lulu’s sisters is wearing a blue ribbon.
Correct solution coming up below.
Hi Kabir,
Here is my solution to #Puzzle 185.1. First let us figure out the numbers. From statements (1), (4) and (5), it is clear that two of the four girls are wearing red ribbons, one is wearing blue and one is wearing orange.
Next, from statements (4) and (5), we can infer that Lulu is not wearing a blue or an orange ribbon, implying that she is certainly wearing a red ribbon.
From statement (1) we can say that Lily is wearing either a blue or an orange ribbon.
Now let us examine statements (2) and (3). Statement (2) implies that the ribbons of Leela's sisters are blue-red-red or blue-red-orange. Statement (3) implies the same thing for Leena, that is to say, the ribbons of Leena's sisters' too are blue-red-red or blue-red-orange. Therefore, the blue ribbon is being worn by neither Leela nor Leena. It means that Lily is certainly wearing the blue ribbon, and between Leela and Leena, one is wearing orange and the other is wearing red, though we can't eliminate either of the two possibilities with the given information.
— Professor Anshul Kumar, New Delhi
***
Only two combinations simultaneously satisfy all the given conditions:
(1) (Lily blue), (Leela red), (Leena orange), (Lulu red)
(2) (Lily blue), (Leela orange), (Leena red), (Lulu red)
What is certain in both combination is that Lily’s ribbon is blue and Lulu’s ribbon is red
What cannot be determined is the colours of Leela and Leena’s ribbons, except that one of them is wearing red and the other one is wearing orange.
— Vinod Mahajan, New Delhi
#Puzzle 185.2
Hi Kabir,
I will start by calling number 1, and then 12, 23, 34, 45, 56, 67, 78, 89, and finally 100 to win the game. Since I want to call number 100 in the end, I need my opponent to call a number between 90 & 99, which means my previous call should be 89. Similarly, for my opponent to call between 79 and 88, I should call 78. And so on, starting with 1 on the first call. When I call 1, my opponent is forced to call a number between 2 and 11, then between 13-22, 24-33, 35-44, 46-55, 57-66, 68-77, 79-88 and finally between 90-99.
— Anil Khanna, Ghaziabad
Solved both puzzles: Professor Anshul Kumar (Delhi), Vinod Mahajan (Delhi), Dr Vivek Jain (Baroda)
Solved #Puzzle 185.2: Anil Khanna (Ghaziabad), Dr Sunita Gupta (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Yadvendra Somra (Sonipat), Ajay Ashok (Delhi), Shishir Gupta (Indore)
