Build paired-digit
spacing sequences
Two of each digit. One row of spaces. Can you give every matching pair exactly the right gap?
Count the digits in between.
The matching digits are the ends of the gap. Don’t count either end as part of the gap.
Here k stands for a digit, such as 1, 2, 3, or 4. Use every supplied tile once. Each digit appears exactly twice. The order of the digits inside a gap does not matter; the number of digits does.
right − left − 1 = gap
So two copies of digit k must be k + 1 positions apart.
Dots are unfinished places, not extra tiles. Every place will contain a digit in the finished sequence.
Try the gap ruler
Choose a digit and move its two positions. Which positions make the gap correct?
Quick checkpoint
A pair of 2s starts in slot 3. Where must the other 2 go to its right?
Give the biggest gap room first.
You have two 1s, two 2s, and two 3s. Build a six-digit sequence that follows all three spacing rules.
Strategy, not a magic rule: starting with the largest gap reduces the choices here. A legal first choice may still need to be changed later.
A dead end is useful information.
Backtracking means undoing a choice that cannot lead to a complete solution, then trying another choice.
Locally correct is not finished.
The 4s can have the right gap, and the 3s can have the right gap, but together they might leave no way to fit the smaller pairs.
Keep a record of the positions you tried. Undoing is part of solving, not a mistake to hide.
What does one failed branch tell us?
Watch for a stronger claim: to say “there is no solution,” you would have to rule out every possible branch, not just one.
Make all the pairs happy.
Try the six-slot puzzle, then switch to the separate eight-slot board, which also includes two 4s. Your six-slot board stays saved. Any arrangement that follows every rule is accepted.
Use two copies each of 1, 2, and 3. Between the two copies of each digit, put exactly that many digits. Fill all six slots.
Slot numbers are positions, not extra digits.
Tap: choose an unused tile, then a slot. Tap a placed tile, then another slot to move or swap it. Placing an unused tray tile onto an occupied slot returns the old tile to the tray. Drag: move a tile directly. Keyboard: focus a slot, press a digit key to place it; Delete removes it. Arrow keys move between slots.
Pair planner · See legal position choices
Choose a pair. These choices fit its spacing rule and do not move any other digit. Clicking a choice places both copies; it may move this pair’s existing tiles. A legal pair does not guarantee a complete solution.
Reference solutions · Reveal only when you are ready
A good solving routine
Plan: start with a large gap and list its possible position pairs.
Try: choose one pair, then update the empty positions.
Check: can the remaining pairs still fit? If not, back up to a choice with an untried alternative.
Verify: every slot is filled, each digit occurs twice, and every pair has the required gap.
One answer… or every answer?
Watch a small search try every legal position pair, largest digit first. It saves each solution, then backtracks to look for more.
Saved solutions
No answers have been found yet. Step through the choices, or finish the search.
Default: read left to right and count reversals separately, as the book’s Practice 5 answers do.
Turn the sequence around
Why does this search find every answer? For each digit, it tries every left-hand slot whose partner is exactly one more than that digit’s value to the right. It skips occupied slots, then repeats the process for the next digit.
After each branch, it undoes that choice and tries the next. Only when all branches have been explored can the display say “Search complete.”
Why does reversing preserve every gap?
The digits between two matching tiles stay between them when the row is reversed. They appear in the opposite order, but there are still just as many.
For example, a pair with two digits inside its gap still has two digits inside its gap after the reversal.
A reversed sequence is therefore another valid left-to-right arrangement. Treating it as a separate answer or as the same mirror pattern is a counting convention, not a change to the spacing rule.
What will you try next time?
Check your understanding. You can change an answer and try again; there is no score penalty.
Four checkpoints complete.
You have checked the gap rule, verified both original puzzles, and explained how to handle a dead end. That is more than finding a lucky arrangement.
My explanation
Which pair did you place first? Why? Describe one choice you changed.
Learning notes & teaching additions
Example 6 supplies two copies each of 1, 2, and 3, with exactly 1, 2, and 3 digits respectively between matching copies. Its printed answer is 312132. Practice 5 extends the rule to two copies each of 1, 2, 3, and 4. The reference section lists 41312432 and 23421314, which are reversals. This lesson therefore counts different left-to-right sequences separately by default.
Added for learning: the gap ruler, the guided position-by-position deductions, the dead-end example, the pair planner, progressive hints, exhaustive search, reversal explanation, and exit ticket. These are teaching additions, not steps printed in the worked example. The search is bounded to two copies of each digit from 1 through the selected maximum, with maximum 1, 2, 3, or 4.
The board checker tests inventory, filled slots, and pair gaps directly. It does not require a student’s row to match the displayed reference answer. A checkpoint means a valid construction has been verified; it does not distinguish independent work from guided work.
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