Construct and Optimize with Digits and Group Sums
用数字构造并优化 · 分组和
Use divisibility, scarce digits, fixed totals, and complete enumeration to build valid arrangements—and then prove that no better arrangement exists.
Conditions first → objective second → proof lastSeparate feasibility, objective, and proof
Construction problems often contain two different jobs: make something legal, then make one measured quantity as large or as small as possible.
Check every printed condition: allowed digits, digit frequency, divisibility, group membership, and total.
Calculate the exact quantity being optimized: the smallest number, the largest number, or a group sum.
Give a bound showing that no arrangement can do better, then give a construction that reaches the bound.
Check the language
Use the last two digits to control divisibility by 4
The worked example construction uses the digits 1 through 9 exactly once to form three three-digit multiples of 4. The last two digits carry the strongest immediate restriction.
Why the four even digits are scarce resources
Every multiple of 4 is even, so each three-digit number needs an even units digit. The only available even digits are:
There are three units places but four even digits. Among three distinct even units digits, at least one is4or8. That ending needs an even tens digit as well, using the fourth even digit. Thus all four even digits are used in the endings, leaving only odd digits for the hundreds places.
All distinct-digit endings from 1–9 that are divisible by 4
Maximize the smallest hundreds digit
After the ending positions are assigned, the three hundreds digits must be odd. To make the smallest number as large as possible, reserve:
Largest possible 5-hundreds candidate
With 7 and 9 reserved as the other hundreds digits, the best remaining tens digit is 8. Divisibility by 4 then forces the units digit 4:
Build and audit your own construction
Use each digit1–9 exactly once across three three-digit multiples of4. Make the smallest of your three numbers as large as possible. To complete this mission, enter any construction attaining584; the order of the three numbers does not matter. The validator checks the printed conditions separately, so a numerical success cannot hide a repeated or missing digit.
Prove the optimum and enumerate every optimum
The worked example proves the upper bound 584 and gives two constructions. A separate runtime search checks every legal construction and determines whether more optimum arrangements exist.
| Construction | Smallest member | Status |
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Minimize the largest two-digit number
original Exercise 4 uses digits 1 through 7 exactly once to form three two-digit numbers and one one-digit number. Their total is100. Minimize the largest of the three two-digit numbers. Enter any valid construction attaining42 to complete this mission; number order does not matter.
Use place value before searching
Let the three tens digits sum to T. The sum of all seven digits is 28.
The three tens digits must therefore be either {1,2,5} or {1,3,4}. To keep the largest number small, use {1,3,4}. The 4-tens number cannot end in 0 or 1, so its least available units digit is 2. This proves a lower bound of 42.
Complete runtime enumeration
Partition 1–8 into three unequal group sums
Partition1–8 into three nonempty groups, using each number exactly once. Find three groups whose sums are all different and whose largest sum is twice the smallest.
Determine the three target sums first
The total is:
Let the ordered sums be:
The inequalities force the integer value:
Then construct actual groups
A sum pattern is not enough. Every number from 1 through 8 must appear exactly once in one group.
Assign every number
Use all six permutations without adding them one by one
Choose three distinct digits from1–9, without repetition. Their six different three-digit permutations total 3330.
Count place-value appearances
Each chosen digit appears:
- twice in the hundreds place;
- twice in the tens place;
- twice in the ones place.
Recover and optimize the digit set
So the three digits must sum to 15. To make the largest permutation as large as possible, maximize the hundreds digit first, then the tens digit. Choose9;the two other digits must total6. Their largest possible distinct pair is5and1, giving951.
Permutation laboratory
State the convention before optimizing a patterned ending
Guided Practice 4 asks for the least n such that 1+2+⋯+n has units digit 3 and tens digit 0. The printed wording permits two readings.
Interpretation A: write a one-digit total as 03
Under a fixed two-character display, the least index is 2.
Interpretation B: require a visible tens place
Under the ordinary numeral interpretation, the least index is 37.
Triangular-ending laboratory
Two-digit numbers equal to four times their digit sum
For a two-digit number 10a+b:
Try it on your own — a new problem
This extra practice does not change your mission score. Try before opening a hint.
Use digits 1,2,3,4 exactly once to make two two-digit numbers. What is the largest possible value of the smaller number?
Independent workshop
Use the strongest structural restriction before attempting a case search. Correct all eight answers to complete this mission.
Exit ticket and certificate
Correct all five answers and complete the other nine missions to unlock the chapter badge.
Certificate of completion
Digit & Group-Sum Optimization Architect
completed Lesson 22.6 by validating constructions, proving extrema, and preserving every mathematically valid optimum.