### 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?

### 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?