"The Operation Combination Problem"

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TL;DR Summary

Education question

"The Operation Combination Problem"

In the "Operation Combination Problem," we explore the space of possible solutions, much like we explore the different possible topologies for the universe. The problem consists of finding all possible combinations of mathematical operations that, when applied to a finite set of integer numbers from 1 to n, produce a given result R.

To solve this problem, we must navigate this "solution space" using the following constraints and specifications:

1. Use only the operations of addition, subtraction, multiplication, and division.
2. Each integer number from 1 to n must be used exactly once.
3. Operations must be performed from left to right, without considering operation precedence (i.e., without applying the "PEMDAS" or "BIDMAS" rule).

By exploring this "solution space," we seek to uncover the different possible "configurations" for combining the integer numbers, just as we seek to understand the different possible topologies of the universe.

For example, if n = 3 and R = 4, one possible combination within this "solution space" might be: (1 × 2) + 3 = 4. This combination represents a specific way of combining the integer numbers from 1 to 3 to achieve the result 4, just as a specific spatial curvature represents a specific way of configuring space in the universe.

SUMMARY:

The problem involves finding all possible ways to combine integer numbers from 1 to n, using only the operations of addition, subtraction, multiplication, and division, to achieve a specific result R. Each integer number must be used exactly once, and operations must be performed from left to right without considering operation precedence.

The goal is to navigate the "solution space" to uncover all possible combinations that yield the result R. The complexity of the problem depends on the chosen values of n and R and can be approached using combinatorial search techniques and solution space exploration.

Side note:

The problem may be of interest to mathematicians and computer scientists, who might propose more efficient approaches to solving it.

 

[anuttarasammyak]

The relation is
[tex](((...(a_1b_1a_2)b_2a_3)b_3a_4)....)b_{n-1}a_n=R[/tex]
where
[tex]1\leq i \leq n, \{a_i\}=\{1,2,3,...,n\} [/tex]
[tex]1\leq j \leq n-1, b_j \in \{+,-,*,/ \}[/tex]
LHS has ##n! 4^{n-1}## patterns. R is a rational number which has equal or less than ##n! 4^{n-1}## different values. We can easily see variety and degeneracy in R by computer.

For an example I assume
[tex] -(n-1)(n-1)!\leq R \leq n!+1[/tex]

 

[pbuk]

By exploring this "solution space," we seek to uncover the different possible "configurations" for combining the integer numbers, just as we seek to understand the different possible topologies of the universe.

Why do you think this is an appropriate analogy?

For example, if n = 3 and R = 4, one possible combination within this "solution space" might be: (1 × 2) + 3 = 4. This combination represents a specific way of combining the integer numbers from 1 to 3 to achieve the result 4

No, it represents a specific way of combining the integer numbers from 1 to 3 to achieve the result 5.

just as a specific spatial curvature represents a specific way of configuring space in the universe.

Again, the analogy is absurd.

The problem may be of interest to mathematicians and computer scientists.

Why do you think that? It might be of some interest to someone for practicing programming techniques, but no more so than the dozens of similar problems on CodeWars.