Move the last digit to the front by multiplication
A 22-digit number ends in 7. Multiply it by 7, and that final digit moves to the front. Every other digit stays in the same order. How can we recover the whole number?
Move the 7.
Keep the block in order.
A digit changes places. The block does not.
Moving a digit is not the same as multiplying. The puzzle asks for a special number for which those two actions give exactly the same result.
Find the hidden 22-digit number
Its first digit is not zero. Digits may repeat, and zeros may appear inside the number.
Remove that final 7 and place it before the remaining 21 digits. Do not reverse, reorder, or remove any of those 21 digits.
Both numbers have 22 digits. Multiplication acts on the whole number, with carries.
original. The book presents the carry method first and the algebraic method second.
Watch a shorter digit move
Predict the moved-digit number before pressing the button.
A quick rule check
Which result keeps all the other digits in the correct order?
Let each carry reveal the next digit
Start with the only known digit: the 7 at the right. Each column tells us what must go immediately to its left.
Why does the next digit become known?
Call the digit just before the final 7 U. The input ends in …U7. After the final 7 moves to the front, the product ends in U.
The units calculation is 7 × 7 = 49. We write 9 and carry 4, so U = 9. We have recovered the ending …97.
The next column uses the carry
Now calculate 9 × 7 + 4 = 67. Write 7, carry 6. That written 7 is the next input digit to the left. The ending is now …797.
The ones digit tells us the next hidden digit. The tens part becomes the carry for the next column.
Do not append 49 or 67 as a block. A single multiplication column writes just one digit.
The 22-column carry workshop
The first 21 columns recover hidden digits. The final column checks that the multiplication closes correctly.
Discover its digits right → left.
Scroll within this panel to see every place. The compact line below always shows the digits you have recovered.
Recover the digit just before 7
The written ones digit must become the next input digit to the left.
All 22 columns fit—including the last one.
The leftmost input digit is 1, with incoming carry 0. The final calculation is 1 × 7 + 0 = 7. It writes the required leading 7 and produces no extra carry. Do not add a 23rd input digit.
Inspect the carry ledger & study controls
This ledger records the columns currently completed in the workshop. A column’s outgoing carry becomes the next column’s incoming carry.
| Column from right | Calculation | Write | New carry | What this tells us |
|---|
Complete a column to start the ledger.
Reveals and replays are learning tools. Checkpoints record a completed check, not whether the work was done without help.
Paper carry workspace
Start at the rightmost 7. In each column, multiply by 7 and add the incoming carry. Write the ones digit; carry the tens part.
| Column from right | Input digit | Incoming carry | Total | Written digit | New carry |
|---|---|---|---|---|---|
| 1 | 7 | 0 | 49 | 9 | 4 |
| 2 | 9 | 4 | |||
| 3 | |||||
| 4 |
Continue through column 22. At the last column, verify that the product writes 7 and produces no extra carry.
Worked solution 1 supplies the long-multiplication construction. The digit ribbons, individual questions, explicit final-column check, and replay are teaching additions.
Turn the digit move into an equation
Instead of recovering 21 digits separately, give the whole hidden block one name: a. It is a 21-digit integer, not a single digit.
Name the hidden block
The first 21 digits form a number. Call that entire number a.
The equation and numerical answer follow worked solution 2. The seven-step pacing and explanatory block diagram are added for this lesson.
Build three important pieces
Fill the blanks. Use the number of digits in the hidden block to choose the exponent.
Why there is at most one answer for these exact settings
The equation becomes 69a = 7 × 1021 − 49. Its right side is fixed, and 69 is not zero. So it determines at most one value of a. We must still check that a is a 21-digit integer and that the resulting number satisfies the puzzle.
This uniqueness explanation is a deduction added to the lesson. It is not a claim that 22 is the shortest possible length; the worked example asks for a 22-digit number.
Let all 22 places agree
Multiply your candidate, move its last digit, and compare the two results. The checker uses the rules—not a memorized answer.
Open the worked solution for study
The book’s two methods produce this same 22-digit number. Revealing it is a study option, not a replacement for checking the multiplication.
Its product by 7 and its moved-digit result are both:
Try the idea with fewer digits
Use the carry strategy on a six-digit challenge, then explore how the last digit, multiplier, and length work together.
End in 4. Multiply by 4.
Find a six-digit number ending in 4 whose product by 4 moves that final 4 to the front. Digits may repeat; the first digit cannot be zero.
The five boxes on the right stay in the same order as the five boxes on the left.
Worked six-digit solution
102564 × 4 = 410256. The zero inside the block 10256 stays in that block when the final 4 moves.
This smaller puzzle and its answer were added for practice. They are not printed on the worked example page.
Change one condition at a time
Choose the final digit d, multiplier m, and total number of digits L. After you try and check the six-digit puzzle, you can review the lab’s complete calculation. Changing the settings alone will not reveal an answer.
Which lengths work for this digit and multiplier?
This finite check is an extension, not a result stated by the worked example. It does not claim anything about lengths above 60.
How the laboratory’s equation is built
Let a be the first L − 1 digits. The original number is 10a + d. The moved-digit number is d × 10L−1 + a. Requiring multiplication by m to do that move gives:
(10m − 1)a = d(10L−1 − m)
The right side must be divisible by 10m − 1 with no remainder. The quotient must be a genuine L − 1 digit number; adding a leading zero does not make a shorter integer longer. Finally, the lab verifies the resulting multiplication and digit move exactly.
The controls use final digits 1–9, multipliers 2–9, and lengths 2–60. These are the laboratory’s exploration limits, not new restrictions on the original 22-digit problem. For each setting the equation supplies at most one candidate, so a failed integer or length check rules out a solution for that setting.
Small columns. One whole-number equation.
Both original methods describe the same multiplication. One looks at a place at a time; the other keeps the leading block together.
✓ Five checkpoints complete
You have followed the digit move, completed the carry construction, built the equation, checked the 22-digit answer, and explained the connection. You have reached the capstone of Chapter 28. Hints and original reveals count as study; these checkpoints are not a test of independent performance.
Turn the rules into something you can check.
A puzzle may ask you to arrange, move, draw, or calculate. First say precisely what must stay true. Then make a useful choice, follow its consequences, and verify the complete result. A successful construction is stronger when you can explain why it works.
Learning notes, exact arithmetic & teaching additions
This is the final unnumbered challenge, not a separately numbered tenth worked example.
The printed task: a 22-digit number ending in 7, multiplied by 7, gives the 22-digit number obtained by taking that final 7 to the front. In this notation, the first 21 digits are A through U. Their order does not change.
Method 1: use long multiplication from low places to high. The first calculation 7 × 7 = 49 determines U = 9 and carry 4. The next calculation 9 × 7 + 4 = 67 determines the preceding digit. The worked example’s completed multiplication gives the number below.
Method 2: let the first 21 digits form the integer a. The worked example writes (10a + 7) × 7 = 7 × 1021 + a, then solves for a.
a = 101 449 275 362 318 840 579
10a + 7 = 1 014 492 753 623 188 405 797
Teaching additions: the short 3407 movement demonstration, detailed carry-column prompts, final-column explanation, replay, staged equation builder, uniqueness explanation for the fixed settings, rule-based checker, near-miss example, six-digit practice puzzle, generalized equation laboratory, bounded length scan, and exit questions. These are identified as explanations or extensions rather than as additional printed original problems.
Place and length conventions: an ordinary L-digit positive integer has a nonzero first digit. Interior zeros and repeated digits are permitted. Moving a digit means moving its single symbol; the other digits keep their order, including their zeros. A leading-zero string is not accepted as an ordinary number of that many digits.
Carry construction: a column uses the current digit, multiplier, and incoming carry. The ones digit of the total becomes the next input digit to the left for the first 21 columns. Column 22 must instead write the required leading 7 and have outgoing carry 0. A repeated digit alone is not a reason to stop.
Checking and bounds: Large integers are calculated using BigInt, never rounded JavaScript Number values. Small digit-and-carry operations use ordinary exact small integers. For interface safety the worked example input accepts at most 40 digits after grouping is removed. The laboratory supports 2–60 digits and single-digit multipliers 2–9. Its length scan checks every length in the displayed range and makes no claim outside that range.
Learning state: optional storage saves this lesson’s workshop position, checkpoints, inputs, quiz responses, and reflection in this browser. No accounts, network requests, external libraries, external fonts, or remote services are used. Work saves automatically when browser storage is available. Reset removes this lesson’s work only. Hints and revealed answers can earn study checkpoints; they do not certify unaided mastery.
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Reference answer and verification
Let a be the first 21 digits. Then 7(10a + 7) = 7 × 1021 + a, so 69a = 7 × 1021 − 49.
a = 101 449 275 362 318 840 579
N = 1 014 492 753 623 188 405 797
7N = 7 101 449 275 362 318 840 579
The last line is exactly the final 7 moved to the front of N.
About this exercise: this lesson, Chapter 28, final “Try It,” /.