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Make 100 with Digit Strings
第21讲 加餐挑战：数字串凑100
Keep every digit in order. At each gap, either join the digits, insert +, or insert −. Build expressions that equal exactly 100.
Learn the exact rules
The challenge is strict. A mathematically correct expression does not count unless it also respects the digit-string rules.
Choose one fixed digit string
Only the eight gaps between neighboring digits may change.
Three choices at every gap
- Join
Leave the gap blank, so adjacent digits become one multi-digit number. - Add
Start a new positive term. - Subtract
Start a new term whose value is subtracted.
A small rule example—not a challenge answer
The blank gap between 1 and 2 joins them into the term 12. Its value is 12−3+4=13. A join changes place value: 12 is not 1+2.
Try the smaller string 234: 2+3+4=9; joining the first gap gives 23+4=27; changing the last sign gives 23−4=19. Notice that replacing +4 by −4 lowers the total by 8, not 4.
Rule checkpoint
On-page hint for Stage 1
A blank gap joins adjacent digits into one number. The digits never change order, and the first number stays positive.
Build and evaluate an expression
Click each gap to cycle through join → + → −. The page evaluates exactly what you built.
On a narrow screen, scroll the digit row sideways. With a keyboard, Tab to a gap and press Enter or Space to change it.
Your discovery notebook
Only expressions you personally submit appear here.
Ascending discoveries 0
Descending discoveries 0
Builder checkpoint
For this stage, make any legal expression that uses at least one join, at least one plus sign, and at least one minus sign. It does not need to equal 100 yet.
On-page hint for Stage 2
Use one gap of each type. This checkpoint checks the controls; reaching 100 is not required yet.
Discover ascending expressions
Use the digits 1 through 9 in order. Find at least two distinct expressions that equal 100.
Type and verify
Goal
Different spacing or typography does not create a new expression. The actual terms and signs must differ.
Discover descending expressions
Now use the digits 9 through 1 in order. Find at least two distinct expressions that equal 100.
Type and verify
Goal
Try organizing cases by the first joined block instead of guessing the eight gaps randomly.
Audit one of your discoveries
A verified expression can be understood as a positive subtotal minus a negative subtotal.
Select a discovered expression
Worked subtotals for my saved discovery
Why this helps
Joined digits determine the terms. The plus and minus signs determine which subtotal each term enters.
The “negative subtotal” is the sum of the magnitudes of all subtracted terms.
Audit checkpoint
On-page hint for Stage 5
Add all positive terms together, then add the magnitudes of the terms being subtracted. Their difference must be 100. With nine digits and k terms, there are 9−k joined gaps.
Prove the search is finite
A systematic search treats the eight gaps independently. Every gap has exactly three possible states.
Three choices in each of eight gaps
With two gaps, the first gap has 3 choices and each of those branches has 3 choices for the second gap: 3×3=9 patterns. Each extra gap triples the count. With eight gaps the count is 3⁸. Each pattern produces exactly one expression for a chosen direction.
Both directions
The ascending and descending strings are two separate search spaces of the same size.
The generated patterns have no duplicates: each sign or join occupies a specific gap. For typed notebook entries, ignore differences caused only by spaces or symbol style.
Pattern odometer
This tool shows one pattern at a time. It never filters directly to the answers.
Search-size checkpoint
On-page hint for Stage 6
Try two gaps first: each of the three choices branches into three more, making nine patterns. Continue multiplying by 3 for each extra gap.
Count every valid expression
This expert extension uses a complete case table. Run every case for both directions, add the case counts, then enter your totals. The computer checks each pattern, but you explain why the cases cover every possibility.
Search nine cases for each direction
Fix the first two gaps. There are 3×3=9 pairs of choices. For each pair, the computer checks all 3⁶=729 choices for the remaining six gaps. Different rows cannot overlap, and together they cover all 6,561 patterns.
Each completed row reports only how many hits it found; it does not show any undiscovered expression. Add the rows to find each direction’s total.
| Case | First gap | Second gap | Ascending hits | Descending hits |
|---|
Why is this a complete search?
Every legal expression has exactly one choice in each of its eight gaps. Its first two choices put it in exactly one row. That row tests every choice for the remaining gaps. Thus no expression is missed or counted twice.
Add and verify the case counts
Optional notebook completeness audit
The badge requires at least two discoveries in each direction plus the completed case-count work. You do not need to save every solution in the notebook to earn it.
The page will answer only “complete” or “not complete yet.”
It will not reveal an undiscovered expression.
Why the count is trustworthy
- Every one of the eight gaps receives one of three states.
- Each generated expression is evaluated with exact integer arithmetic.
- Only expressions totaling 100 are counted.
- Canonical formatting prevents duplicates.
Explain completeness and claim the badge
The written reflection is for your own reasoning. It is not graded by keywords or length.
Final search checklist
Reflection (not automatically graded)
Make 100 Digit-String Enumerator
Bonus explorer
You preserved digit order, discovered valid constructions, audited their arithmetic, and verified a complete finite search.
Chapter 21 optional challenge
On-page hint for Stage 8
Explain which row contains an arbitrary expression and why all choices for its remaining gaps are tested. A large number of trials alone does not prove completeness.
Learning notes
This page preserves those rules but does not reproduce the printed sample answers.The complete-count stage is an added exhaustive-search extension generated directly from the rules.