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Base Puzzle

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 .Octi the octopus counts.

Happy Octopus

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.

Basically

The number 3723(in base 10) is written as 123 in another base. What is that base?

Oh for the Mathematics of Yesteryear

Age 11 to 14
Challenge Level

Correct solutions were received from Mary (Birchwood Community High School) and Chor Kiang. Well done.

Part 1

First, find the total ounces of bread.
$600 \times35 \times24$ ounces $= 504000$ ounces

Now, find the new number of men-shares needed.
$4800 \times45 = 216000$

Divide $504000$ ounces by $216000$ to get $2\frac{1}{3}$ ounces a day.

Hence, each man must only eat $2\frac{1}{3}$ ounces of bread a day.

Part 2

$ £14  10\text{s} = 290\text{s}$

$ £5 \; 8\text{s } 9 \text{d} = 108.75$ shillings
 

Weight (cwt)   Distance (miles) Amount (shillings)
$60$ can be transported for $20$ for $290$
$1$ can be transported for $20$ for $\frac{290}{60} = \frac{29}{6}$
$1$ can be transported for $30$ $\frac{29}{6}\times\frac{30}{20}= 7.25$
$\frac{1}{7.25}$ can be transported for $30$ $1$
$\frac{1}{7.25}\times 108.75$ can be transported for $30$ $108.75$

Since $\frac{108.75}{7.25} = 15$

$15$ cwt can be carried for $30$ miles at a cost of $ £5 \; 8\text{s } 9 \text{d}$