Matrix and Missing Number Grids

Overview

Matrix and Missing Number Grids appears about 1 times per paper across the last four AFCAT solved papers, placing it in the high yield band of Reasoning and Aptitude.

A matrix or missing-number grid item in AFCAT shows you a small box of numbers (most often 3 rows by 3 columns, sometimes 4 by 3 or 3 by 4) where one cell is replaced by a question mark. Your job is to look at the cells you can see, work out the rule that links them, and use that rule to find the value that fits the empty cell.

The exam gives you about one item of this kind in a typical paper, so it is a single mark — but it is a mark you can win quickly if you have a clean checking order in your head. The trouble for most candidates is not the arithmetic; the numbers are small. The trouble is wandering through possible rules at random and running out of time. This page fixes that by giving you a fixed order to test rules, a short list of the rules AFCAT actually uses, and a clear catalogue of the traps the paper-setter likes to set.

The same method works for letter grids and mixed letter-number grids, which are rare in AFCAT but do turn up in model papers and other defence exams you may sit alongside it. Once you have the rule-testing habit, every grid item — whatever the surface dressing — collapses to the same short routine.

Why matrix grid items reward rule extraction

A matrix item is not really an arithmetic question. The arithmetic in the cells is almost always primary-school level — small whole numbers, easy products, simple squares. What the item is actually testing is whether you can spot a hidden rule from a small sample of evidence.

That is exactly the skill the AFCAT reasoning section is built around. The paper-setter shows you two complete rows (or two complete columns) and one incomplete row. The two complete rows are your training data. They tell you which operation the setter has in mind. The third row is the test — you apply the rule you just learned and read off the answer.

This framing matters because it tells you where to spend your time. Do not stare at the missing cell. Stare at the two complete rows. The moment you have nailed down a rule that works for both of them, the missing-cell calculation is one line of arithmetic. Most candidates who get these wrong do so because they jumped to the missing cell first and tried to guess a rule that made the answer look nice — instead of proving the rule on the rows where they had full information.

Common grid sizes and the standard test-rule order

AFCAT grids come in three layouts. Each layout suggests where to look first.

Whichever layout you see, run the test-rule order in this fixed sequence:

1. Rows first. Take the top row and ask, does column 3 equal column 1 plus column 2? If not, try column 1 times column 2. Then column 2 minus column 1. Move down to test row 2 with the same rule. If both rows confirm the rule, apply it to the row with the missing cell.
2. Columns next. Only if no row rule works. Ask the same questions vertically — does the bottom cell equal the top plus the middle?
3. Diagonals last. If neither rows nor columns yield a rule, look at the main diagonal (top-left to bottom-right) and the anti-diagonal (top-right to bottom-left).

Following this order saves time because the most common AFCAT grid is row-driven. You will solve roughly four out of five items before you ever look at columns.

Standard rules to test

The AFCAT paper-setter draws from a short, repeating list of rules. Memorise this list and run through it in order. The rules below are written for a three-column row with cells a, b, c where c is the third column you are predicting from a and b.

If none of these eight rules fits, you are probably looking at a multi-step rule (the next section) or a column or diagonal rule rather than a row rule.

The rule-discovery method, step by step

The method below is the exact routine to run on the page in the exam hall. It uses two known rows to prove the rule and the third row only at the end.

1. Pick the two rows that are complete. Call them row 1 and row 2. The row with the missing cell is row 3.
2. List the candidate rule. Start with sum across — write a + b and check whether it equals c in row 1.
3. Check the same rule in row 2. A rule must work in both complete rows. If sum works in row 1 but not row 2, discard sum and move to product.
4. If the simple rule fails by a constant factor, try sum and multiply. For instance, if a + b = 6 but c = 18 in row 1, and a + b = 4 with c = 12 in row 2, the rule is (a + b) times 3.
5. Apply the proven rule to row 3. Substitute the two known cells, compute, and read off the missing cell.

Cross-row interactions vs within-row rules

Most AFCAT grids use a within-row rule — every row is self-contained, and the rule is the same in each row. A small number use a cross-row interaction, where the cell in row 3 is built from cells in rows 1 and 2 of the same column.

You can spot a cross-row pattern quickly. Compare row 3 column by column with the cells directly above. If row 3 column 1 looks like a clean function of row 1 column 1 and row 2 column 1 (most often their sum, difference, or product), you are dealing with a cross-row rule.

When you have established that the rule runs down columns, the same eight standard rules apply — sum down, product down, difference down, and so on. The mental setup is identical; only the direction has changed.

A useful default — if no within-row rule fits the two complete rows, do not waste time inventing exotic row rules. Switch direction. Within-row rules and cross-row rules are mutually exclusive in AFCAT; the setter picks one or the other, not both.

Multi-step rules — two operations chained

A small but real number of AFCAT grids use a rule that needs two operations rather than one. These are still made of the standard ingredients, just joined.

The rule-discovery method still works. Run sum and product first; if both fail by a clean factor (the same factor in both complete rows), you are looking at a multi-step rule. The factor itself is the second operation.

Letter-matrix grids

Letter grids look different, but the underlying logic is identical. Each letter has a hidden number — its position in the alphabet, where A is 1, B is 2, and so on through Z which is 26. The rule that links the cells is then an arithmetic relation among these positions.

The routine is simple. First, convert each letter you can see into its alphabet position number. Second, run the standard rule-discovery method on the numbers. Third, convert the answer number back into a letter.

For example, if the grid shows B, C, E across the top and the rule is sum across (1 plus 2 plus 2 equals 5 for B, C, E because B equals 2, C equals 3, and E equals 5, with the second number — 3 — being the gap between B and E rather than the middle cell), check the second row in the same way. The conversion is mechanical; the reasoning is exactly the same as for a number grid.

AFCAT keeps letter grids rare, but the same technique handles them when they appear.

Mixed letter-number grids

A mixed grid has some cells with numbers and some with letters. The rule typically links the number cell to the letter cell through the alphabet-position conversion.

For instance, a row might read 3, C, 9 where the rule is column 1 equals the alphabet position of column 2, and column 3 equals column 1 times 3. Or a row might read B, 4, 6 where the alphabet position of column 1 plus column 2 equals column 3 (2 plus 4 equals 6).

Three quick checks handle most mixed grids:

• Convert every letter to its position number. Write the numbers above the letters on your rough paper.
• Treat the grid as fully numeric. Run the standard rule-discovery method.
• Convert the answer back to a letter if the missing cell is a letter.

The conversion adds about ten seconds. The reasoning load is the same as for any number grid.

Grid items with diagonal rules

If no row rule and no column rule fits the two complete rows, the rule almost certainly runs along a diagonal. There are two diagonals in a 3 by 3 grid — the main diagonal from top-left to bottom-right, and the anti-diagonal from top-right to bottom-left.

Diagonal rules in AFCAT are usually one of three forms:

• Sum along the diagonal. The three diagonal cells add up to a constant — often the same as the row sum.
• Multiplication along the diagonal. The product of the three diagonal cells equals a constant.
• Mirror sum. Each cell on the main diagonal plus the cell at its mirror position on the anti-diagonal gives the same total.

To test a diagonal rule, calculate the sum (or product) of both diagonals from the cells you can see. If both diagonals give the same total or product, you have your rule, and the missing cell becomes the value that completes the matching total along its diagonal.

Common AFCAT trap patterns

The grid item is a low-difficulty mark in theory but a high-error mark in practice. The setter relies on a few standard traps to catch candidates who rush.

Two-rule items where the rule changes between rows

Two-rule grids are rare in AFCAT but you should know how to spot them. In a two-rule grid, row 1 follows one operation (say sum) and row 2 follows a different operation (say product). The missing cell sits in row 3, which uses yet another rule — and the trick is that the three rules together form a pattern.

The most common version is the operation-rotation grid — row 1 is sum, row 2 is product, row 3 is difference (or another permutation of the same three operations). To handle these, do not test for a single consistent rule. Instead, identify the rule for row 1 alone, the rule for row 2 alone, and then look at what is left of the standard list — that is the rule for row 3.

A second version is the multiplier-progression grid — row 1 uses (a + b) times 2, row 2 uses (a + b) times 3, row 3 uses (a + b) times 4. The base operation is the same, but the multiplier increases by 1 down the rows.

If you see two-rule patterns in a model paper or coaching test, the giveaway is that no single rule from the standard list fits both complete rows. Switch to the two-rule frame only after you have exhausted the single-rule possibilities — most grids stay single-rule.

Time budget for grid items

AFCAT gives you 2 hours for 100 questions, or 72 seconds per question on average. The grid item is single-mark, single-step once you have the rule, so it should not eat into your average.

That gives you a total of about 50 to 55 seconds — comfortably inside the 72-second average. If you have spent 60 seconds and still have not found a rule, the item is probably a column or diagonal grid in disguise; flag it, move on, and come back at the end with a fresh head. Spending 90 seconds on a single-mark item costs you a chance to bank two or three other marks.

What AFCAT grid items actually look like

Most AFCAT grid items follow one of four broad shapes:

• Pure sum-across grids. The simplest form — column 3 is column 1 plus column 2 throughout. Often used in model papers as a confidence-builder.
• Pure product-across grids. Column 3 is column 1 times column 2. Numbers stay small so the products remain manageable.
• Sum-and-multiply grids. Column 3 is (column 1 plus column 2) times a constant. This is the single most common harder shape on AFCAT.
• Power or square grids. Column 3 is the square of column 1 plus column 2, or some similar combination. Numbers in column 3 are noticeably bigger than in columns 1 and 2.

Recognising the shape from the size and spread of the visible numbers cuts your rule-testing time in half. If column 3 is much bigger than columns 1 and 2, do not bother testing sum or difference — go straight to product, power, or square.

Worked AFCAT-style examples

Exam-day strategy

1. Redraw the grid on rough paper as a clean 3 by 3 (or 4 by 3) box before testing any rule — misreading the layout is the most common error.
2. Test rules in a fixed order — rows first, then columns, then diagonals. Within rows, try sum, product, difference, then sum-and-multiply.
3. Always prove the rule on both complete rows before applying it to the row with the missing cell. A rule that fits only one row is a coincidence, not a rule.
4. If column 3 is much larger than columns 1 and 2, skip sum and difference and jump straight to product, sum of squares, or square of sum.
5. If the simple sum is too small but the gap is the same factor in both complete rows, you are dealing with a sum-and-multiply rule.
6. Convert letters to alphabet positions (A is 1, Z is 26) as soon as you see a letter grid. Treat the rest as a number grid.
7. Aim for 50 to 55 seconds. If you pass 60 seconds without a rule, flag the item and move on; revisit at the end with a fresh head.

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