### Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

### Nim

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

### Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

##### Stage: 3, 4 and 5

We are accustomed to writing numbers in base ten, using the symbols for 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, 75 means 7 tens and five units. However numbers can be written in any number base.

If we use base 8 instead of base ten, then 75 is written as 113 which denotes one sixty-four (82 ), one eight (81 ) and 3 units (instead of hundreds, tens and units).

Base 2 is particularly useful as it only requires two symbols, for zero and one, and it is the way numbers are represented in computers.

Just as, in base ten, the columns represent powers of 10 and have 'place value' 1, 10, 102 , 103 etc. (reading from right to left), so in base 2, the columns represent powers of 2. Hence the number 1001011 denotes (reading from right to left):

1 unit (20 ), 1 two (21 ), no fours (22 ), 1 eight (23 ), no sixteens (24 ), no thirty-twos (25 ), 1 sixty- four (26 ).

The number 1001011 in base 2 is the same as the number 75 in base ten.

As another example, we use the symbols 0, 1, 2, 3 and 4 to represent numbers in base 5. The columns in base 5 have 'place value' 1, 5, 25, 125, 625 etc reading from right to left. The number 75 in base ten is the same as the number 300 in base five, that is 3 twenty-fives, no fives and no units.

The number 4102 in base 5 denotes 2 units, no fives, 1 twenty-five and 4 one hundred and twenty-fives making altogether 527 in base ten.

Writing the number 75 in base six we get 203, which represents 2 thirty-sixes, no sixes and 3 units.

We have seen that 75 (base10), 1001011 (base 2), 300 (base 5), 113 (base 8), and 203 (base 6) all represent the same number.

Similarly, we can write 75 in any base we choose and we can write all numbers in any base.

To write numbers between 0 and 1, we use negative powers of the base. For example, in base 2 we use halves, quarters, eighths, sixteenths etc instead of the tenths, hundredths, thousandths etc. which we use in base ten.

So if we write 11.11 in base 2 this denotes $2^1 + 2^0 + 2^{-1} + 2^{-2}$. The equivalent in base 10 is $2 + 1 + {1\over 2} + {1\over 4}$, that is 3.75 in base 10.