HI Malik,

I would do it in two steps. First convert the number you have in base 6 to base 10, and then convert the base 10 expression to base 4. For example let's look at $1023_6$ which is read five three one base 6. To express this number in base 10 think of what $1023_6$ means.

\[1023_6 = 1 \times 6^3 + 0 \times 6^3 + 2 \times 6 + 3 = 1\times 216 + 0 \times 36 + 2 \times 6 + 3 = 216+ 12 + 3 =231.\]

Thus $1023_6 = 231_{10} = 231.$

To express 231 in base 4 you repeatedly divide by 4 and record the remainders. Here is the table I used to keep track.

• I divided 231 by 4 and got 57 with a remainder of 3
  
• I divided 57 by 4 and got 14 with a remainder of 1
  
• I divided 14 by 4 and got 3 with a remainder of 2
  
• I divided 3 by 4 and got 0 with a remainder of 3

Reading the remainders from the bottom to the top I see 3213. Thus 231 is $3213_{4}.$

Hence $1023_6 = 3213_{4}.$

If this is not clear think of it this way. Suppose you have 231 marbles and some paper bags. Arrange the marbles by putting 4 in each bag and you have 57 bags of 4 marbles each and 3 remaining marbles.

You also have some cardboard boxes, each box large enough to hold 4 bags of marbles. Arranging them this way you have 14 boxes with 4 bags each and 1 bag remaining.

Now arrange the boxes into stacks of 4 boxes each and you have 3 stacks with 2 boxes remaining.

Hence you have 3 stacks, 2 boxes, 1 bag and 3 marbles which is $3 \times 4^3 + 2 \times 4^2 + 1 \times 4 + 3$ marbles $=3213_4$ marbles.

I hope this helps,
Penny