Answers to “Can you solve these puzzles?”

Unit Octagon

Name the regions as follows:

$E$ is a 1×1 square. Placed together, $A$, $C$, $G$ and $I$ also make a 1×1 square. $B$ is equal to $H$ and $D$ is equal to $F$.
Therefore $B+E+F=A+C+D+G+H+I$. Therefore the hatched region is $C$ larger than the shaded region. The area of $C$ (and therefore the difference) is $\frac{1}{4}$.

Reverse Bases

If $ab$ in base 10 is equal to $ba$ in base 4, then $10a+b=4b+a$. So, $9a=3b$. $a$ and $b$ must both be less than 4, as they are digits used in base 4, so $a=1$ and $b=3$. So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:

• 23 in base 10 is equal to 32 in base 7.
• 46 in base 10 is equal to 64 in base 7.
• 12 in base 9 is equal to 21 in base 5.
• 24 in base 9 is equal to 42 in base 5.

Unit Octagon

Name the regions as follows:

$E$ is a 1×1 square. Placed together, $A$, $C$, $G$ and $I$ also make a 1×1 square. $B$ is equal to $H$ and $D$ is equal to $F$.
Therefore $B+E+F=A+C+D+G+H+I$. Therefore the hatched region is $C$ larger than the shaded region. The area of $C$ (and therefore the difference) is $\frac{1}{4}$.

Reverse Bases

If $ab$ in base 10 is equal to $ba$ in base 4, then $10a+b=4b+a$. So, $9a=3b$. $a$ and $b$ must both be less than 4, as they are digits used in base 4, so $a=1$ and $b=3$. So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:

• 23 in base 10 is equal to 32 in base 7.
• 46 in base 10 is equal to 64 in base 7.
• 12 in base 9 is equal to 21 in base 5.
• 24 in base 9 is equal to 42 in base 5.

Matthew is a postdoctoral researcher at the University of Cambridge. He hasn’t had time to play Klax since the noughties, but he’s pretty sure that Coke is it!

• My favourite LaTeX package

The Chalkdust editors share some of their favourites
• On the cover: cellular automata

Discover the meaning of the coloured squares on the cover of issue 13
• Oπnions: Should I share my code?

Scroggs debates whether sharing truly is caring
• Crossnumber winners, issue 11

Did you solve it?
• Chalkdust issue 11 puzzle hunt #1

Matthew Scroggs sets the first puzzle. Can you solve it?
• Crossnumber winners, issue 10

Did you solve it?