Problem
Find numbers meeting digit clues
A number crossword. Each Across and Down clue describes a number by digit conditions. Where an Across and a Down cross, the digit must match. Work out each clue and settle the open ones by their crossings.
Reasoning · Grade 2-2 Conditions and Numbers
Your answer
Watch out A clue read alone often leaves several candidates. The digit a crossing hands you settles the rest.
How to solve
Strategy Guess and Check — Each clue is a small constrained search: I list the numbers that fit the rule (Make a Systematic List), then use the crossing cells to Eliminate the ones that clash with a neighbor, leaving the one value that fits both directions. Clues like A4, A5, D1, D2 are forced by their own definitions.
1STEP 1
Clues forced by their own definition
Write in 9319, 19, 1009 and 8990, the clues their descriptions already fix.
A4: 9,3,1,9 = 9319; A5 = 19; D1 = 1009; D2 = 8990
Each of these clues nails down exactly one number, so they anchor the grid before the crossings even matter.
3.NBT.A.2Guess And Check2STEP 2
A1 forced by the D1 crossing
The leading 1 of 1009 fixes the first digit, so that Across run is 147.
A1 ∈ {147, 258, 369}, first digit = 1 → 147
The grid hands us the leading digit for free, which singles out the one +3-pattern number that starts with 1.
3.OA.D.9Eliminate Possibilities3STEP 3
D5 then A6, forced by the A5 = 19 crossing
A5's 1 ends the 4-cell Down run, so it is 4321; its 3 then fixes that Across as 7531.
D5 ∈ {3210, …, 9876}, ones = 1 → 4321; A6 ∈ {6420, 7531, 8642, 9753}, tens = 3 → 7531
One digit travels: A5's 1 pins down D5, and D5's 3 then pins down A6 — each pattern list has exactly one survivor.
3.OA.D.9Eliminate PossibilitiesOne known digit travels through a shared cell and leaves exactly one survivor in the next answer's list.
Why?
A cell where two runs cross belongs to both at once, so whatever digit one answer puts there the other answer must accept.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Every candidate whose digit clashes at that cell is crossed out, and when only one candidate is left it has to be the answer.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
4STEP 4
D3 and D4, hanging from the 9s in A4 and A5
Under A4's leading 9 the Down run is two cells, so it is 91; under A5's 9 it is three cells, all the same, so 999.
D3 = 9□, 9 + 1 = 10 → 91; D4 = 9□□ (all same) → 999
Both Down answers hang from a 9 that an already-solved Across put in place, and the length of the run does the rest of the work.
2.OA.B.2Make A Systematic List5STEP 5
A2 and A3 from the digits the Down answers left behind
The palindrome takes 9 from D1 and 8 from D2, so it is 9889; and 0 + 1 = 1 + 0 makes the last Across 1001.
A2 = 9 8 □ □ (palindrome) → 9889; A3: 0 + 1 = 1 = thousands + tens, thousands ≠ 0 → 1001
Once the Down answers have dropped a digit into each remaining Across, the clue's own rule fills the empty cells with no choices left.
3.OA.D.9Eliminate PossibilitiesAnswer
across 147, 9889, 1001, 9319, 19, 7531 / down 1009, 8990, 91, 999, 4321
Every answer satisfies its own rule AND every shared cell: A1 = 147 starts with D1's 1; A5 = 19 leaves a 1 at the foot of D5, so D5 = 4321, and D5's 3 is A6's tens digit, giving A6 = 7531; D3 = 91 fills the two cells under A4's leading 9 (9 + 1 = 10, odd); D4 = 999 fills the three cells under A5's 9; A2 = 9889 takes 9 from D1's last cell and 8 from D2's first cell; A3 = 1001 takes 0 from D2's last cell and 1 from D3's last cell, and 0 + 1 = 1 + 0. A4 = 9319, A5 = 19, D1 = 1009, D2 = 8990 each match their own definition exactly. All shared cells agree, so the filled numbers are consistent.
Takeaway
Solve the easy clues first (A5 = 19, D1 = 1009), then let the shared cells hand you a digit so a pattern clue like A6 has only one number that fits — 7531!
- Clues forced by their own definition
- A1 forced by the D1 crossing
- D5 then A6, forced by the A5 = 19 crossing
- D3 and D4, hanging from the 9s in A4 and A5
- A2 and A3 from the digits the Down answers left behind