Problem # 1: "The Itinerant Monk"

The legendary Zen monk Nue-Toan was very fond of visiting hilltop shrines.

On a lovely Fall day in 1729 BC, exactly at sunrise, he started on the path that would take him up the hill to the shrine of Lord Kor-Tezan. He stopped occasionally for a drink of pure spring water and to take deep breaths of the clean mountain air. He reached the hilltop exactly at sunset.

He spent the evening and night at the shrine, meditating and sleeping (not that an observer could easily tell the difference). The next morning, exactly at sunrise, he left the shrine and started down the hill on the same path. Again, he stopped occasionally, sometimes to drink and eat, and other times to simply take in the spectacular scenery. He reached the base of the hill at sunset.

Here's the question: prove that somewhere along the path from base to hilltop, there is a place that Nue-Toan passed by AT EXACTLY the same time on both days. You don't have to identify the place or the time - simply convince me that such a place and time exist.

Problem # 2

What is the largest number you can write using 3 digits. You may use a digit more than once.

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Please send your solutions to both problems to anil@lokvani.com.

Use "Problem Solutions M-032703" as the subject line. Please include your full name in the text of the main message. The first one to submit the right answer will be profiled in the next issue of Lokvani.

Please do not post your solution in "Post Comments". No credit will be give for solutions not sent to anil@lokvani.com.

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SOLUTIONS TO PROBLEMS ON 2/27/03

Problem # 1

We have a bunch of rectangular blocks and we also have a chessboard. Each rectangular block is exactly the size of two adjacent squares on the chessboard. So the chessboard can be fully covered with 32 of these blocks.

Now suppose we cut off two corner squares of the chessboard (as shown in the picture). Is it possible to cover this new board (now consisting of just 62 squares) with 31 blocks?

Solution

No, it is not possible to cover the chopped off board with 31 blocks.

The key insight here is the realization that the two corner squares that were chopped off ARE OF THE SAME COLOR. Therefore, the chopped off board has 32 squares of one color and 30 squares of the other color.

Since every rectangular block covers one black square and one white square, 31 blocks will cover 31 white squares and 31 black squares. But alas, there are only 30 squares of one of the two colors. So it is impossible to cover the chopped off board with 31 blocks.

Extra credit comment :-)

Here's something neat and subtle:
Let's say the original puzzle statement had used an 8 by 8 square (as opposed to a chessboard). Obviously now there is no built-in notion of white and black squares. But we can imagine overlaying a black-white alternating pattern, solve the problem as described above, and then mentally "remove" the pattern. So, in a sense, we could have "invented" the chessboard to solve the real problem and then gotten rid of it when we were done.

Problem # 2

A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.

What are the exact costs of each item?

Solution

The costs of each item are: $3.16, $1.50, $1.25, and $1.20.

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