Rules of the Year Game

  • Rules

    Use the digits in the year 2021 and the operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and !! (double factorial) along with grouping symbols, to write expressions for the counting numbers 1 through 100.

    1. All four digits must be used in the expression.
    2. Only the digits 2, 0, 2, 1 may be used.
    3. Multi-digit numbers such as 20, 210, or 0.22 will be accepted this year.  
    4. The square function may NOT be used. Nor may the cube, raise to a fourth power, or any other function that raises a number to a specific power. For example, (2 + 0)^2 - 1! is an acceptable way to write 3, because ^ is an acceptable operation and it uses exactly the digits 2, 0, 2, and 1. But 2^2 + 0! + 2 + 1! is not an acceptable way to write 8, because "^2" is not an acceptable operation, and there are not three 2's available. Similarly, 2^3 + 0! - 2 - 1! would not be an acceptable way to write 6 since "^3" is not an acceptable operation.
    5. Multifactorials other than the double factorials may NOT be used this year. Although explored in previous years, many people felt they took away some of the challenge, and they were very hard to read. 
    6. The integer function may NOT be used. Nor may the round, floor, ceiling, or truncate functions.
    7. Remember that 0! by definition is equal to 1. 
    8. Depending on the context where 0^0 occurs, it can be 1, indeterminate, or undefined/nonexistent. For this game we will accept the value 0^0=1. 

    The four digits can be used in any order, but we encourage you to challenge yourself to use the digits in the order 2, 0, 2, 1. We also encourage solutions using only the single digit numbers 2, 0, 2, and 1 rather than double digits like 0.2 or 22.