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Newsletter puzzles

These puzzles appeared in the Chalkdust newsletters. You can find the answers here and sign up for the newsletter here.

121

Source: Mathematical Circus by Martin Gardner
Find a number base other than 10 in which 121 is a square number. 

Making the most of your ice cream cone

You have a circular piece of paper with which to construct two hollow cones. You cut out a sector with an angle $\theta$ then glue together the straight edges of the sector. You can make the other one by stickgin together the straight edges of the residual paper.

Which angle $\theta$ should you choose in order to maximise the volumes of the two cones? 

Polya Strikes Out

Source: Thinking Mathematically by John Mason, Leone Burton & Kaye Stacey
Write the numbers 1, 2, 3, … in a row. Strike out every third number beginning with the third. Write down the cumulative sums of what remains:

1, 2, 3, 4, 5, 6, 7, …
1, 2, 3, 4, 5, 6, 7, …
1, 2, 4, 5, 7, …
1=1; 1+2=3; 1+2+4=7; 1+2+4+5=12; 1+2+4+5+7=19; …
1, 3, 7, 12, 19, …

Now strike out every second number beginning with the second. Write down the cumulative sums of what remains. What is the final sequence? Why do you get this sequence? 

These puzzles appeared in the Chalkdust newsletters. You can find the answers here and sign up for the newsletter here.

121

Source: Mathematical Circus by Martin Gardner
Find a number base other than 10 in which 121 is a square number. 

Making the most of your ice cream cone

You have a circular piece of paper with which to construct two hollow cones. You cut out a sector with an angle $\theta$ then glue together the straight edges of the sector. You can make the other one by stickgin together the straight edges of the residual paper.

Which angle $\theta$ should you choose in order to maximise the volumes of the two cones? 

Polya Strikes Out

Source: Thinking Mathematically by John Mason, Leone Burton & Kaye Stacey
Write the numbers 1, 2, 3, … in a row. Strike out every third number beginning with the third. Write down the cumulative sums of what remains:

1, 2, 3, 4, 5, 6, 7, …
1, 2, 3, 4, 5, 6, 7, …
1, 2, 4, 5, 7, …
1=1; 1+2=3; 1+2+4=7; 1+2+4+5=12; 1+2+4+5+7=19; …
1, 3, 7, 12, 19, …

Now strike out every second number beginning with the second. Write down the cumulative sums of what remains. What is the final sequence? Why do you get this sequence? 

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