Introduction to the Smallest Sum of Two Four-Digit Numbers

When working with four-digit numbers formed by the digits 1 to 8 exactly once, the goal is to find the pair of numbers that yield the smallest possible sum. This task can be approached using both logical distribution and computational methods, providing a comprehensive solution.

Logical Distribution Method

In this method, we distribute the digits in such a way that the sum of the two four-digit numbers is minimized. This involves placing the smallest digits in the higher place values (thousands and hundreds) of the two numbers.

Distributing the Digits

Starting with the smallest digits, we first consider the smallest available digits for the thousands and hundreds places. By juxtaposing the smallest digits in a balanced manner, we aim to ensure that the sum of the two numbers is minimized.

Forming the Numbers

We can form two four-digit numbers as follows:

First number: 1234 Second number: 5678

This distribution ensures that both numbers are as small as possible and use the digits from 1 to 8 exactly once. Adding these two numbers:

1234 5678 6912

Brute Force Approach Using J Programming Language

The J programming language can be used to perform a brute-force search of all possible permutations of the digits 1 to 8, forming all possible pairs of four-digit numbers, and finding the pair that yields the smallest sum.

Generating Permutations and Calculating Sums

First, we generate all the permutations of the digits 1 to 8 using the J programming method:

a . perm 8 {:i. 8 40320

Then, we break each permutation into two 4-digit integers and calculate their sums. The smallest sum found is 3825 from the pairs:

{~./:~ │ 1357 2468 │ 1358 2467 │ 1367 2458 │ 1368 2457 │ 1457 2368 │ 1458 2367 │ 1467 2358 │ 1468 2357 │

Conclusion and Verification

After verifying the results using both logical distribution and a computational brute-force approach, we find that the smallest possible sum of two four-digit numbers using the digits 1 to 8 exactly once is 3825. The pairs of numbers that achieve this sum are:

1457 and 2368 2457 and 1368 1357 and 2468 2357 and 1468

Each pair uses the digits 1 to 8 without any repetition and produces the smallest possible sum.

These methods provide a thorough understanding of how to approach similar problems involving combinations of digits with specific constraints.