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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

old wooden military catapult.A garrison of $600$ men has just enough bread to allow $24$ ounces a day to each man for $35$ days; but, with the news that the enemy was planning an attack, the garrison was forced to reinforce to $4 800$ men. The enemy arrives and decides to enforce a siege - to reduce their own loss of life. How many ounces of bread a day must each man in the garrison be allowed, to hold out $45$ days against the siege of the enemy?

If the transport of $60$ cwt. (hundredweights) of bread for $20$ miles costs the garrison $ £14 \; 10\text{s}$. (i.e. Fourteen pounds ten shillings), what weight can be carried $30$ miles for $ £5 \; 8\text{s } 9 \text{d}$?

n.b. in those days:

$12$d(pennies) = $1$s (shilling)

$20$s = $ £1$ (pound)

$16$ ounces = $1$ lb (pound - "weight")

$112$ lb= $1$ cwt