Problems of the Week

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2017-09-04 Basic


Ned is trying to play music using a steel ruler extending beyond the table edge. When he strikes the ruler, a twang sound is produced.

If he wants to produce a higher frequency note, then what should he do?

A pen costs an integer number of cents. There are 100 cents in a dollar.

9 of these pens cost between $11 and $12.
13 of these pens cost between $15 and $16.

What is the price of one pen in cents?

A filament bulb is connected with a battery, a resistor, and two switches \(S_1\) and \(S_2,\) as shown in the diagram above. For which combination of the switches will the bulb glow?

Assume the connecting wires have negligible resistance.

\[\begin{array} { r c r } 3^3 &=& {\color{red}{2}}7 \\ 3^4 &=& {\color{red}{8}}1 \\ 3^5 &=& 2{\color{red}{4}}3\\ 3^6 &=& 7{\color{red}{2}}9\\ 3^7 &=& 21{\color{red}{8}}7 \\ 3^8 &=& 65{\color{red}{6}}1 \end{array} \]

The above shows the first few powers of 3, starting from \(3^3\). Is it true that the second last digit (from the right) of \(3^n\) for integers \(n\,(> 2)\) is always an even number?

There is a knight on the bottom left square of a \(101\times 101\) chess board. What is the minimum number of moves required for the knight to get to the top right corner?

Note: A knight's valid moves are shown by the stars in the picture to the right.


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