One-position change puzzle: find the shortest route

The One-position change puzzle here can have many successful routes and still have a clear minimum number of moves. In the original paper exercise here, the start is ABCA and the target is CBBC. One move replaces the letter in exactly one position. We will show why three moves are necessary, then write a route that reaches the target in three.

Use four fixed places and only the letters A, B and C. Repeated letters are allowed. The rows do not have to be dictionary words, and the letters carry no hidden message. These are invented rules for a pencil exercise, not a commercial game, a security system or a method for handling a real access code.

Write the permitted move before trying a route

Draw four spaces across a working page and number the positions one through four above them. Copy ABCA into the first row and CBBC into a separate target row. Keep both visible. Each row always contains four letters in the same four places, read from left to right. A position is a place in the row, rather than the name of a particular letter.

An allowed move selects one position and replaces its letter with a different member of A, B and C. You may not insert a fifth letter, delete a place, exchange two positions, or slide a letter along the row. Choosing the same letter again makes no replacement and does not count as a move. Write the whole resulting row after each actual move.

This rule lets any position receive its target letter directly. There is no restriction requiring a letter to pass through another value first. That freedom matters to the minimum we will prove. A puzzle with extra restrictions would need its own analysis; the answer below belongs to these stated rules.

Compare positions rather than bags of letters

Read the first position of ABCA beside the first position of CBBC. A differs from C, so that place needs a change. At position two, both rows have B. At position three, C differs from B. At position four, A differs from C. There are three mismatched positions: one, three and four.

The two appearances of A in the start occupy separate places. A change at position one leaves position four untouched. Likewise, the target's two B letters are not interchangeable physical pieces that can be picked up and redistributed. We are replacing entries in fixed positions. Counting the different letter names would answer another question.

To make this visible, put a small mark beneath each mismatched column on your working page, keeping the mark outside the four-letter row. Leave the matching second column unmarked. Those three marks describe the starting comparison. They are not three moves already completed, and they do not yet show a route.

Prove that two moves cannot be enough

Each of the three initially wrong positions must change at least once before the final row can equal the target. If position one is never selected, its original A remains there and the result cannot begin with C. The same reasoning applies independently to positions three and four. Every successful route therefore has to touch all three.

One permitted move touches only one position. Two moves can touch at most two different positions, even if you choose them carefully. They cannot fix all three initially mismatched places. This gives a lower bound of three moves: any successful route needs at least that many.

The argument does not assume that every move is sensible. Someone might change a matching position, change a wrong position to another wrong letter, or revisit the same place. Those choices still touch only one position per move. They cannot make two moves cover the three separate places that require a change.

Keep the phrase “at least” beside this result. A lower bound rules out shorter routes. By itself, it does not demonstrate that a route of exactly three moves is available. We still need to write one under the same rules rather than treating the count as a completed solution.

Build a route that meets the bound

Start at ABCA. For the first move, replace the A in position one with C. The new row is CBCA. The second position remains B, the third remains C, and the fourth remains A. Only the selected first entry has changed.

For the second move, replace the C in position three with B. The row becomes CBBA. For the third move, replace the A in position four with C. The row becomes CBBC, exactly the target. The complete written route is ABCA, CBCA, CBBA, CBBC.

There are four rows because the starting row is included. There are three moves between them. Read each adjoining pair and identify its one changed position: first, third, fourth. This check establishes that the displayed route uses permitted moves and actually reaches the specified target.

Now join the two parts of the answer. The position argument showed that fewer than three moves cannot work. The displayed route succeeds in three. Together they prove that the minimum is three. Neither a confident label saying “shortest” nor a successful route on its own would supply both parts.

A longer successful route is still longer

Try the invented route ABCA, BBCA, CBCA, CBBA, CBBC. Its first move changes position one from A to B. Its second changes that same position from B to C. The last two moves correct positions three and four. Every step is permitted, and the final row is correct, but the route contains four moves.

That extra B was an unnecessary intermediate value under our rules. Position one was allowed to become C directly. The four-move route demonstrates reachability, not minimality. Keep it on a separate comparison line if you want to show why arriving at the right answer does not settle how few moves were needed.

Do not repair the longer route by calling its first two replacements one action. The movement rule counts one actual replacement per move. Combining two written rows under a bracket would change the presentation, not the number of replacements made. The shorter route avoids the intermediate value instead.

A shortest route need not be the only route

You can correct the three mismatched positions in another order. For example, ABCA can become ABCC by changing position four, then CBCC by changing position one, and finally CBBC by changing position three. This is another three-move route to the same target.

Both routes meet the lower bound. Their coexistence does not weaken the minimum result, because the question asks for the smallest number of moves, rather than a uniquely determined order. Label the result “minimum three” without adding a claim that only one route works.

The freedom to choose the order comes from the exercise's independent positions. Changing one entry does not move another entry or prevent another place from receiving its target letter. We have not imposed a rule about using a limited stock of physical letter tiles. Such a rule would create a different problem.

Carry the reasoning to another starting row

For a second invented example, compare ABBA with ABBC under the same rules. Only position four differs. One replacement changes its A to C, so the minimum is one. A long visual resemblance between rows is not the argument; the explicit matching and mismatching positions supply it.

If the start already equals the target, no replacement is needed. Leaving the row alone gives a zero-move route. The rule that a move must change a letter does not require you to make a move when the task is already complete. Keep that case separate from pretending that a no-change gesture counts as an allowed replacement.

More generally within this exact exercise, count the positions that differ from the target. Each must change, and one move can correct only one. Then set each mismatched position directly to its target letter. This produces a route with the counted number of moves. The lower bound and construction agree because no other restrictions have been added.

Keep related puzzles with their own rules

The adjacent card-swap puzzle changes an order by exchanging neighboring cards. Its minimum and parity arguments concern those exchanges. This exercise replaces letters in fixed places; borrowing the swap count would answer the wrong movement question.

The eraser smearing guide addresses an actual changed mark, the intended correction area and tool-care limits. The paper proof here supplies no erasing technique and does not diagnose a drawing. You can keep a fresh written row instead of physically removing a previous proposal.

Read the actual writing supplies separately

The Mars rectangular block eraser offer identifies a white latex-free eraser and describes individual cellophane wrapping. Its gallery shows STAEDTLER Mars plastic blocks in a display box. The title's outer bundle of ten boxes is not established by that one package photograph. No erasing trial is reported here, and the description's ink-from-film wording is not transferred to paper.

The BAZIC pink eraser-top offer describes caps for most standard pencils. Its gallery shows pink hollow wedge-shaped caps and a twelve-pack label, while the title states an outer total of 3456. Those are different packaging levels. The exercise establishes no actual pencil fit, delivered count, universal safety or clean-result guarantee.

Separate a supply order from the paper result

If this pencil exercise leads you to consider writing supplies on republeq.com, keep the proposed order separate from the mathematical answer. A minimum of three replacements says nothing about how many erasers a particular bundle supplies or whether an eraser cap fits a pencil you own. Resolve the actual item and packaging questions before deciding to buy. If the current seller offers a cryptocurrency route and you want to use USDT, read the instructions for that identified transaction rather than borrowing details from another order. Confirm the requested asset, network, recipient and amount before authorizing a transfer. This paragraph does not establish token availability for either linked offer, supply a payment address, or report a completed purchase. A familiar product name is not enough to identify an unchanged basket if the version or quantity has changed. Keep the item reference with any seller reply, then check that the order information still describes what you chose. If important information remains unclear, leave the purchase pending while you obtain it; the paper exercise can be completed without an order. Read the actual delivery, cancellation, return and refund terms for the payment route you are considering. Do not infer reversibility, refund denomination or an arrival date from the word cryptocurrency. Retain whatever confirmation the real transaction produces, describing only what it actually confirms. That record concerns the purchase. The worked rows concern an invented puzzle. Neither record supplies missing facts for the other, and a successful payment cannot prove the minimum-move argument or a product's effect on a particular sheet.

Leave both parts of the answer on the page

For ABCA to CBBC, retain the three mismatched positions beside a three-move route. Another reader can inspect why fewer moves fail and how the stated minimum is attained. Keep the fixed positions, alphabet and one-replacement rule with that result. Changing those rules starts a new puzzle rather than silently extending this answer.