page_title: Nimber
section_title: Addition and multiplication tables
hierarchical_section_title: Nimber / Addition and multiplication tables
context_page_description: In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
  Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.
  Nimbers have the characteristic that their Left and Right options are identical, following a certain schema, and that they are their own negatives, such that a positive ordinal may be added to another positive ordinal using nimber addition to find an ordinal of a lower value. The minimum excludant operation is applied to sets of nimbers.
context_section_description: The following tables exhibit addition and multiplication among the first 16 nimbers. 
  
  This subset is closed under both operations, since 16 is of the form 2²ⁿ.
  (If you prefer simple text tables, they are here.)
caption: Nimber multiplication of powers of two (sequence A223541 in the OEIS)
  Calculating the nim-products of powers of two is a decisive point in the recursive algorithm of nimber-multiplication.