Escape the Number System Vault
Five doors. Four bases. One way out.
Your mission
You're locked in a vault with five doors, and each one opens only with the right number in the right base. First master the three conversion tools, then try them on your own numbers, then crack every door to escape.
Explore
Open each of the three tools and step through its example to the end.
Every vault door is locked with a number in a different base. To escape, you need three tools. Choose a tool above and step through how it works.
0 of 3 tools learned
Experiment
Convert your own numbers between any two bases. Try at least three conversions, using at least two different tools.
Choose two bases, type a number and press Convert. The workbench shows every step.
0 of 3 conversions
Challenge
Type the right number to open each door. Get one wrong, and the vault shows you how it works before you try again.
Open all 5 doors to continue.
Understand
Here is what the vault was teaching you, in the words you need for the exam.
Which tool for which job?
| Converting | Use | How |
|---|---|---|
| Binary, octal or hex to decimal | Place values | Multiply each digit by its place value (a power of the base), then add. |
| Decimal to binary, octal or hex | Repeated division | Divide by the base again and again. Read the remainders from the bottom up. |
| Binary to octal or hex | Bit grouping | Split into groups of 3 (octal) or 4 (hex) from the right. Pad with zeros on the left. |
| Octal or hex to binary | Bit grouping | Replace each digit with its 3-bit or 4-bit group. |
| Octal to hex (or back) | Go through binary | Expand to binary, then regroup. |
Place values
Octal 347 = 3 × 8² + 4 × 8¹ + 7 × 8⁰ = 192 + 32 + 7 = 231.
Hex 2D7 = 2 × 256 + 13 × 16 + 7 × 1 = 727. Remember: A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15.
Repeated division
156 ÷ 8 = 19 remainder 4
19 ÷ 8 = 2 remainder 3
2 ÷ 8 = 0 remainder 2
Read the remainders upward: decimal 156 = octal 234. Stop when the answer reaches 0.
Why grouping works
3 bits make 8 patterns, exactly the 8 octal digits. 4 bits make 16 patterns, exactly the 16 hex digits. So each group swaps for one digit. Binary 110101111 becomes 0001 1010 1111, which is 1AF in hex.
Exam tip
The two classic mistakes: reading remainders from the top down (it's bottom up!), and padding on the right when grouping (always pad on the left, because zeros on the left don't change the value). Check every answer by converting it back.