How to Solve Greater-Than Sudoku on Paper with a Pen
NetSudoku Editorial Team
Reading time 7 min ยท Technique Guide
Table of Contents โผ
A classic board hands you two dozen digits and asks you to reconcile them. A Greater-Than board hands you almost none โ its hints are not digits at all but a network of < and > signs hung between every adjacent pair of cells. The first time you see one in print, the grid looks like a set of arithmetic exercises someone forgot to finish.
On paper this is a gift, not a punishment. Because there is no undo button, the variant rewards one specific habit the classic puzzle never teaches: treating the board as a graph and walking it. Before any digit is written, the sign structure alone pins sweeping candidate ranges โ and unlike classic Sudoku, where you earn placements through scanning, these placements fall out of chain length. The method below is the shared pen-and-paper Series Hub workflow bent around a single new object: the red arrow.
The two-pen rule: finals in blue, logic noise in red. Every arrow you draw is red; every struck candidate is red. Blue ink appears only when a cell's candidate range has collapsed to one digit it provably must take.
1. The Rules, Stated Precisely for Paper
- Classic Sudoku holds: every row, column, and 3x3 box contains digits 1โ9 exactly once.
- Some adjacent cell pairs are joined by an inequality sign. When two cells A and B carry A
>B, the digit in A must be strictly larger than the digit in B. - The sign is read from the larger cell to the smaller cell โ the open mouth always faces the bigger digit.
- An edge with no sign has no restriction. This matters: Greater-Than boards typically mark most, not all, interior edges.
- Every puzzle is unique-solution; a valid grid must satisfy every embedded inequality simultaneously.
One number dominates the whole strategy: because the inequality chain is transitive, five arrows strung in one direction consume the digits below their head. A chain of length n forces the head cell to be at least n+1. Memorise that single relationship and the variant stops looking random.
2. Paper Adaptation โ Re-Draw Every Sign as a Red Arrow
Printed Greater-Than signs sit in a pixel-sized gap between cells, so the eye cannot read them as structure. The entire game changes when you re-render them:
With your red pen, squeeze a small arrow โบ into every signed gap, always pointing from the larger cell toward the smaller cell. After one sweep the board reads as a directed graph: arrows visibly stream downhill. Do this before copying any givens โ the arrows are the givens.
Then mark the extremes: circle every cell that only emits arrows (its neighbours are all smaller) and every cell that only receives arrows (its neighbours are all larger). These cells are your first blue-ink commitments โ a pure emitter can never be anything but a low digit, a pure receiver nothing but a high one.
3. Pass 1 โ Endpoint Seizure, 1s and 9s
The cheapest wins in Greater-Than are topological. A cell with out-pointing arrows on both sides โ it must beat its left neighbour and its right neighbour โ is already confined; add a second constraint and it collapses to 9.
The 1-Seizure
Any cell whose arrows all point into it beats nothing โ every neighbour outranks it. Since digits are positive, its value is bounded below by neither; combined with the chain math (see Pass 2) it usually dies to 1. Write the 1 in blue and strike 1 from every cell its arrows touch: those neighbours are now 2โ9, and any of them that sees six larger neighbours climbs to 2 minimum.
The 9-Seizure
A cell with two or more out-pointing arrows was at minimum 2. When its row, column, or box already contains a 2, 3, 4, 5, 6, 7, and 8 โ or when the reverse chain above it shops six distinct smaller digits โ it must be 9. Seizing a 9 is the strongest move in the variant because it deletes a candidate from every peer cell across both houses.
4. Pass 2 โ Walk the Chains, Pin the Ranges
Take each maximal run of arrows pointing the same way, from the high end to the low end, and write a shrinking range down the chain. A cell sitting at the head of a three-arrow cascade beats three distinct smaller digits, so its own digit is at least 4; walk down and each successor drops the floor by one.
For a chain A โบ B โบ C โบ D the four cells must hold four distinct descending digits. The minimum possible is 4,3,2,1, so A โฅ 4 regardless of everything else; the maximum possible is 9,8,7,6, so D โค 6. Write these floors and ceilings as candidate ranges in the corner of each cell โ that one sweep resolves whole rows on medium boards.
Keep a mental clock on the arithmetic: every digit committed abandons its range, and the ranges of cells along the chain shift toward their forced floor or ceiling. A cell whose floor and ceiling meet at one number is solved โ write it in blue and let the collapse ripple.
5. Pass 3 โ Cross-Hatch the Corners and Merge Houses
By the third pass the arrow data on the board's interior is exhausted and the remaining work is classic Sudoku on a board of ranges. Cross-hatch the corner boxes first: their three edge-adjacent neighbours carry the weakest sign load, so row, column, and box scans shine there. Then render every range as a candidate set and reduce with naked pairs, hidden pairs, and box/line interactions exactly as you would on a plain grid โ see the hidden pairs page and the naked pairs page when a house stalls.
The merge rule is simple: a cell's final candidates are the intersection of its arrow range with row-digit-strikes, column-digit-strikes, and box-digit-strikes. When an intersection member is knocked out by an arrow downgrade, strike it with one red line โ never erase โ so the audit trail of "why" stays on the page.
6. The Extreme Chain, Drawn the Way You Will Draw It
The strip reads exactly as your board should after the red-arrow pass. Five arrows stream in one direction; the head cell outranks five distinct neighbours and the tail cell is outranked by five โ chain math pins both extremes, and one house strike closes each.
Walk it left to right: the leftmost cell emits five arrows, so it must beat five distinct digits, and a single column strike removes 6, 7, 8 โ leaving 9. Its neighbour therefore starts at best at 8 but must still sit below the locked 9, then each step down the chain loses one more ceiling; the tail cell, beaten by all five, is forced to 1 the moment its column drops 2 through 8. Ranges collapse into givens, and the 1 and 9 propagate into the two neighbouring boxes.
7. The Four Moves That Decisively Break Greater-Than
7.1 Minimum / Maximum Extremes
The transitive core of the variant. Any cell with a chain of n arrows pointing one way at its tail can be pinned to a floor of n+1; run the mirror arithmetic for ceilings. When a floor and a ceiling meet, the pair of constraints all by itself resolves the cell โ no digit work required. This is why Greater-Than solvers count arrows before they count candidates.
7.2 Chain Walking
A run of 2 < 3 < 4 in a row forces the highest digit to the rightmost cell and the lowest to the leftmost within the chain's footprint. Treat a chain as a single sliding unit: the width of the run constrains where the unit can sit inside its row, and any digit placed outside the unit tightens the unit's internal order. Walk chains before you touch pairs โ they carry an order guarantee pairs never offer.
7.3 Endpoint Seizure of 1s and 9s
Pure emitters and pure receivers are the variant's only instant-resolves. A cell with out-pointing arrows on both sides (left and right, or up and down) holds a middle-to-high digit automatically; a cell with in-pointing arrows on both sides holds a low digit. Once either is trapped against a second house, it becomes a hard 9 or 1 โ always write those in blue immediately, then let the neighbour ranges absorb the shock.
7.4 Cross-Hatching the Corners
Corner boxes collect warped arrow data, so classic cross-hatching carries the late game: sweep a digit across the three houses of a corner box exactly as in the beginner tips drill. Because ranges already whittled most candidates, a hidden single found this way usually terminates a whole corner.
8. Common Paper Mistakes on Inequality Boards
- Reading the sign backwards. On paper it is embarrassingly easy to transfer a printed
<into a red arrow flipped the wrong way. Verify every arrow against its neighbour once before you trust the graph. - Ignoring unmarked edges. A missing sign is a free edge, not a clue โ but it is also not a logjam. When a range sits in an unsigned gap, both dispositions are still legal; do not strike either until a house forces it.
- Committing the head of an over-long chain. A nine-cell downhill chain trivially forces its head to 9 โ but a five-cell chain only forces the head to be โฅ 6. Assume nothing beyond the exact run length you counted.
- Skipping the arrow audit. A board whose final digits contradict an embedded sign is failed by definition. Spend the last sixty seconds walking every red arrow pair-to-pair.
9. Train the Graph Eye, Then Carry It to Paper
Greater-Than rewards range intuition that carries across the whole family. Rehearse on NetSudoku's classic boards first โ start at Easy, then Medium, Hard, and Expert โ translating every candidate hunt into a fake chain walk: "what range would a five-arrow run pin here?" When classic boards feel slow, re-read the strategy index and the X-Wing technique guide before returning to ink.
Greater-Than is the cleanest gate into transitive reasoning, and every arrow trick you learn here transfers to Consecutive Sudoku, where marked edges replace inequalities with the i-and-i+1 relation. Walk the chains, seize the ends, and the board's skeleton does the heavy lifting.
Validate Your Arrow Work on a Real Grid
Every NetSudoku puzzle is generated with a verified unique solution โ solve the same board on paper, then compare your directed graph against the answer key.