Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Base Puzzle

## You may also like

### Happy Octopus

### Oh for the Mathematics of Yesteryear

Or search by topic

Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions

1000, ?, 100, 31, 24, 22, 20, 17, 16, 15, 14, 13, 12, 11, 10

The solution to this problem is that all the numbers are the decimal number 16 in bases ranging from base 2 (binary) to base 16 (hexadecimal).

The first number, 10000, is in base 2, corresponding to 16 in decimal or base ten: with components one 16 and no 8's, 4's, 2's or 1's,

Therefore the second number should be in base 3, with components one 9, two 3's and one unit. The missing number is 121.

Solutions were received from Alex Skilton, age 15, The King's School, Canterbury, UK, Soh Yong Sheng, and Ling Xiang Ning of Raffles Institution, Singapore, and David Lowe, age 15, Trinity School, Carlisle, UK.

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.

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... 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?