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 S1S_1 and S2,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.

33=2734=8135=24336=72937=218738=6561\begin{array} { r c r } 3^3 &=& {\color{#D61F06}{2}}7 \\ 3^4 &=& {\color{#D61F06}{8}}1 \\ 3^5 &=& 2{\color{#D61F06}{4}}3\\ 3^6 &=& 7{\color{#D61F06}{2}}9\\ 3^7 &=& 21{\color{#D61F06}{8}}7 \\ 3^8 &=& 65{\color{#D61F06}{6}}1 \end{array}

The above shows the first few powers of 3, starting from 333^3. Is it true that the second last digit (from the right) of 3n3^n for integers n(>2)n\,(> 2) is always an even number?

There is a knight on the bottom left square of a 101×101101\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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