Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers 
  
    
      
        X
      
    
    {\displaystyle X}
  
, 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 and 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
, each containing 
  
    
      
        k
      
    
    {\displaystyle k}
  
 elements, and a bound 
  
    
      
        b
      
    
    {\displaystyle b}
  
. The goal is to select a subset 
  
    
      
        M
      
    
    {\displaystyle M}
  
 of 
  
    
      
        X
        ×
        Y
        ×
        Z
      
    
    {\displaystyle X\times Y\times Z}
  
 such that every integer in 
  
    
      
        X
      
    
    {\displaystyle X}
  
, 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 and 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 occurs exactly once and that for every triple 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle (x,y,z)}
  
 in the subset 
  
    
      
        x
        +
        y
        +
        z
        =
        b
      
    
    {\displaystyle x+y+z=b}
  
 holds.
This problem is labeled as [SP16] in.
 Take 
  
    
      
        X
        =
        {
        3
        ,
        4
        ,
        4
        }
      
    
    {\displaystyle X=\{3,4,4\}}
  
, 
  
    
      
        Y
        =
        {
        1
        ,
        4
        ,
        6
        }
      
    
    {\displaystyle Y=\{1,4,6\}}
  
 and 
  
    
      
        Z
        =
        {
        1
        ,
        2
        ,
        5
        }
      
    
    {\displaystyle Z=\{1,2,5\}}
  
, and 
  
    
      
        b
        =
        10
      
    
    {\displaystyle b=10}
  
. This instance has a solution, namely 
  
    
      
        {
        (
        3
        ,
        6
        ,
        1
        )
        ,
        (
        4
        ,
        4
        ,
        2
        )
        ,
        (
        4
        ,
        1
        ,
        5
        )
        }
      
    
    {\displaystyle \{(3,6,1),(4,4,2),(4,1,5)\}}
  
. Note that each triple sums to 
  
    
      
        b
        =
        10
      
    
    {\displaystyle b=10}
  
. The set 
  
    
      
        {
        (
        3
        ,
        6
        ,
        1
        )
        ,
        (
        3
        ,
        4
        ,
        2
        )
        ,
        (
        4
        ,
        1
        ,
        5
        )
        }
      
    
    {\displaystyle \{(3,6,1),(3,4,2),(4,1,5)\}}
  
 is not a solution for several reasons: not every number is used (a 
  
    
      
        4
        ∈
        X
      
    
    {\displaystyle 4\in X}
  
 is missing), a number is used too often (the 
  
    
      
        3
        ∈
        X
      
    
    {\displaystyle 3\in X}
  
) and not every triple sums to 
  
    
      
        b
      
    
    {\displaystyle b}
  
 (since 
  
    
      
        3
        +
        4
        +
        2
        =
        9
        ≠
        b
        =
        10
      
    
    {\displaystyle 3+4+2=9\neq b=10}
  
). However, there is at least one solution to this problem, which is the property we are interested in with decision problems.
If we would take 
  
    
      
        b
        =
        11
      
    
    {\displaystyle b=11}
  
 for the same 
  
    
      
        X
      
    
    {\displaystyle X}
  
, 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 and 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
, this problem would have no solution (all numbers sum to 
  
    
      
        30
      
    
    {\displaystyle 30}
  
, which is not equal to 
  
    
      
        k
        ⋅
        b
        =
        33
      
    
    {\displaystyle k\cdot b=33}
  
 in this case).
 Every instance of the Numerical 3-dimensional matching problem is an instance of both the 3-partition problem, and the 3-dimensional matching problem.
 NP-completeness of the 3-partition problem is stated by Garey and Johnson in "Computers and Intractability; A Guide to the Theory of NP-Completeness", which references to this problem with the code [SP16]. It is done by a reduction from 3-dimensional matching via 4-partition.
To prove NP-completeness of the numerical 3-dimensional matching, the proof is similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used.