Writing numbers of one system in another system
To convert a number from a system whose base is not 10 into decimal, write it in expanded form and compute the sum: 111010₂ = 1 · 2⁵ + 1 · 2⁴ + 1 · 2³ + 0 · 2² + 1 · 2¹ + 0 · 2⁰ = 32 + 16 + 8 + 2 = 58. To convert a whole decimal number into base p, divide it by p with remainder, divide the quotient by p again, and go on until the quotient is less than p. The last quotient and all the remainders (starting from the last, i.e. from bottom to top) give the digits of the required number: for 100 : 2 the remainders are 0, 0, 1, 0, 0, 1 and the last quotient is 1; read from bottom to top, 1100100. If a remainder is above 9, a letter is written (10 = A, 11 = B, …). When moving between two other systems, the decimal system is a “bridge”: first to decimal, then to the required base. Between systems with bases 2, 4, 8, 16 there is a shortcut: one octal digit matches a group of 3 binary digits (triad), one hexadecimal digit matches a group of 4 (tetrad), and one base-4 digit matches a group of 2 (dyad). For example 3C₁₆ = 0011 1100₂.
“Chain of remainders”: each team gets a decimal number. Pupils take turns at the board doing one division and writing the remainder; at the end the remainders are read from bottom to top. Teams with the right answer score points.