Inverse position map: restore the original card order

Declare what each position number means

Inverse position map is a return list for a declared rearrangement. Start this original paper exercise with five cards in the order A, B, C, D, E. Number their original positions one through five from left to right. The rearrangement list is 3,5,1,2,4. Its first entry means read the card from original position three first; its second means read original position five next. Continue through the whole list to create a new row. These are invented card positions, not product identifiers or instructions for a device.

The direction of that rule matters. We are listing where each new position reads from, rather than listing a destination for each original card. Under this read-from convention, the original card C goes into the first new position because three is the first entry. Write the convention beside the numbers before calculating a return list. A correct list interpreted in the opposite direction can describe a different rearrangement.

Build the shuffled row from the complete list

Follow the five entries one at a time. Position three contains C, so C comes first. Position five contains E, so E comes second. Position one supplies A, position two supplies B, and position four supplies D. The shuffled row is therefore C, E, A, B, D. Each original position has been selected once. No card has been inserted, removed or copied in this declared operation.

Keep the original row, the read-from list and the shuffled row in separate records. The letters identify the card values in this example, while the numbers identify positions. A note containing only C, E, A, B, D would show the result but omit the rule that produced it. Here the rule is supplied explicitly. We are constructing its inverse, without claiming that any final row by itself reveals the history of every action that led to it.

Locate each original index inside the first list

To put A back first, find where original index one appears in 3,5,1,2,4. It appears in the third entry. The shuffled row consequently holds A in position three, so the first entry of the return list is three. Next find original index two. It appears in the fourth entry, making four the second return entry. Check those two locations against C, E, A, B, D: reading its third and fourth positions gives A and B.

Original index three appears in the first entry, original index four in the fifth, and original index five in the second. Append those locations in original-index order. The complete return list is 3,4,1,5,2. This is the inverse position map under our declared convention. Every entry answers the same question: where in the shuffled row is the card from this next original position? Mixing that question with where a card will move afterward would change the list's meaning halfway through.

Read the shuffled row using the return list

Apply 3,4,1,5,2 to C, E, A, B, D as another read-from list. Its third position supplies A; its fourth supplies B; its first supplies C; its fifth supplies D; and its second supplies E. The result is A, B, C, D, E, exactly the original row. Verify all five positions rather than stopping after A and B agree. A list can begin correctly and still exchange later cards.

Also check that the return list itself uses positions one through five once each. Its entries are three, four, one, five and two, with no repetition or missing index. Thus it rearranges the current row without copying or discarding a position. The restored letters demonstrate the chosen example, while the complete-position check confirms that the written return instructions meet the kind of operation this exercise permits.

Keep the two lists in different roles

Call the first read-from list P and the return list Q. These names distinguish their jobs without changing their entries. In this example P is 3,5,1,2,4 and Q is 3,4,1,5,2. To check the first return entry, take Q's three and look at P's third entry, which is one. For the second return entry, Q gives four and P's fourth entry is two. Continuing this check returns the original indices one through five in order.

This is a useful check when copying lists into a worksheet. It operates on the position records, so it does not depend on how attractive or familiar the card labels look. Read a return entry as a position in P, then inspect the original index stored there. The result should be the index whose return entry you were checking. If the third result is five instead of three, inspect the third return entry before declaring the whole procedure wrong.

Explain why locating positions constructs an inverse

For a general row with n positions, suppose P contains each index from one through n exactly once. The shuffled value in position i is the original value in position P's ith entry. To construct Q's jth entry, locate the unique position in P that contains j. That location exists because j appears, and it is unique because j appears only once. The construction provides one return entry for every original position.

Now restore position j by reading the shuffled row at that Q entry. The first rearrangement put the original position j there, precisely because P contained j at that location. Reading it back therefore returns the original jth value. The same reasoning applies to every j from one through n. This establishes restoration for any such position permutation and any original row values, rather than relying only on the five printed letters in our worked example.

Do not replace the inverse with a familiar reversal

Reversing the shuffled letters C, E, A, B, D produces D, B, A, E, C. That is a valid reversal of the row, but it is not A, B, C, D, E. The original operation was not defined as a reversal, so reversing its result has no general reason to undo it. A reversal can undo another reversal when the same complete units are used; this example instead needs the return list built from P.

Repeating P is another different proposal. Applying 3,5,1,2,4 again to C, E, A, B, D produces A, D, C, E, B. Some positions happen to return, but the whole row does not. Do not accept the proposal because its first letter agrees. An operation can be its own inverse in some cases, yet that property must be checked for the actual list. Here the explicit second application shows that P is not its own return list.

Reject a repeated index before promising a return

Consider a changed first list, 3,5,1,1,4. It selects position one twice and never selects position two. Applied to the original A, B, C, D, E row, it produces C, E, A, A, D. There is no B to recover by selecting positions from that output. The lookup rule also fails directly: original index two has no location in the changed list, while original index one has two locations. It is not a permutation of the five position IDs.

Keep that changed input separate from the successful P example. You could consult the saved original and write a fresh B, but writing a new value is another operation, not an inverse rearrangement of the existing output. The complete-and-unique index requirement prevents this loss before a return list is promised. It does not authorize guessing a missing entry merely to make the first list look like the one that worked earlier.

Distinguish repeated printed values from repeated indices

If the original printed labels are A, A, B, C, D but the same valid P is still supplied, the position map continues to have its inverse Q. The shuffled row becomes B, D, A, A, C. Reading its positions 3,4,1,5,2 restores A, A, B, C, D. The two identical printed As do not remove a position from the known map. Positions one and two remain distinct positions even when their displayed values agree.

Without the map or separate occurrence identities, identical printed labels may leave their physical history unresolved. A final row showing two As does not identify which particular A traveled through which route. Keep this narrower uncertainty separate from the result we proved. A supplied permutation can be inverted for repeated values; recovering an unknown history from indistinguishable marks is a different question.

Bound the optional workspace accounts

The Begin Home Decor horse desk-pad account shows flat and rolled views of a horse-print pad in one composite image. Those poses do not verify two supplied pads, its exact maker or model, dimensions, material or protection performance.

The Begin Home Decor tree desk-pad account shows flat and rolled views of a tree-print pad. Exact maker, model, dimensions and material remain unverified. No grip, cleaning or protection-performance claim is adopted.

Neither account supplies the cards, position lists or inverse proof. Their role is optional workspace context, and the exercise requires no purchase.

Leave neighboring order questions distinct

The castanet card order guide counts inversions and checks minimum and exact lengths for adjacent swaps. This exercise supplies a complete read-from permutation and constructs its inverse. It sets no move budget or restriction to neighboring exchanges.

The reversible line poem guide preserves complete sentences while reading their units in reverse order. Here the selected rearrangement is not a reversal, and the task is to recover an original row with a separately constructed return list.

Keep any order enquiry outside the card positions

If you separately consider a desk-pad order on republeq.com, assess the current offer and transaction before choosing how to pay. The card positions in this worksheet are invented mathematical inputs, not product quantities, prices or cryptocurrency instructions. Before using USDT, inspect the actual checkout and determine what that particular order presents. Check the requested amount and denomination, the receiving destination displayed, any network instruction, and the order status before sending funds. These are questions to resolve, rather than statements that an option exists for either offer. Confirm that the item and intended quantity match your proposed order. Resolve product and supplied-content questions separately from payment instructions. A photograph does not authenticate transaction terms. Keep the current instructions, their date and the result of any action you actually take, distinguishing a pending step from a completed payment. If the selection changes while you are checking, review the resulting transaction instead of carrying an earlier conclusion across automatically. Determine what the merchant currently supplies about delivery, cancellation and any refund process before relying on it. Do not assume a transfer can be reversed or a particular network will be supported. This conditional checklist makes no claim about token acceptance, transfer success or available remedies. You can complete the paper worksheet without buying either offer. A restored card row cannot authenticate a payment destination or show that an order has been accepted. The actual purchase needs its own clear and current order-specific evidence, and its records belong separately from this abstract exercise. Retain unanswered questions beside the proposed transaction.

Save the convention with the two complete rows

Keep A, B, C, D, E beside P = 3,5,1,2,4 and its output C, E, A, B, D. Retain Q = 3,4,1,5,2 with the restored row and the location check for each original index. The complete record shows what was rearranged, how each new position read its value and why the return list works. If the first list changes, rebuild its inverse from its actual entries instead of carrying this example's Q into a different operation.