This work brings universal and upside-down magic squares of order 3 to 16. The work is for two digits as {1,8}, {2,5} and 6,9. It can easily to extended for the digits {0,1} and {0,8}. In case of orders 8, 9 and 16 the bimagic squares are also constructed. The block-wise constructions for the orders 8, 9, 12, 15 and 16 are also done. In case of order 15, two ways are given. One as blocks of pandiagonal magic squares of order 5 and another as semi-magic squares of order 3. In both the cases, the magic square of order 15 is semi-magic. The extension of this work to magic squares of orders 17 to 32 is given in another paper. The whole work is without use of any programming language. It is just based on the combination of numbers.