Coding-decoding: Reasoning Ability Notes for Duolingo English Test

Coding‑Decoding Reasoning

Purpose
Coding‑decoding questions test your ability to recognise a pattern or rule that maps a set of symbols (the “code”) to another set of symbols (the “decoded” representation).
The key is to find the consistent rule that works for all given pairs and then apply it to a new pair.

1. Simple Explanation

Identify the pairs – You’re given at least two coded‑decoded pairs.
Look for patterns –
- Numbers → letters (A=1, B=2, …)
- Letters → numbers (Vowel=1, Consonant=2)
- Arithmetic operations (add, subtract, multiply)
- Positional changes (shift, reverse, alternate)
Test your hypothesis – Apply the rule to the first pair; if it works, test it on the second.
Apply to the test pair – Once the rule is confirmed, decode the new code.

2. Key Points to Remember

#	Point	Why It Matters
1	One rule only	Coding‑decoding uses a single consistent rule, not multiple different rules for each pair.
2	Check both directions	The rule can map code → decoded or decoded → code; decide the direction from the context.
3	Look for simple operations first	+, –, ×, ÷, or +digit/‑digit are most common.
4	Consider letter positions	A=1, B=2, … Z=26; sometimes the rule involves reversing or shifting positions.
5	Check for alternation	Even/odd positions may follow different sub‑rules.
6	Use the given answer choices	They often hint at the rule (e.g., if one choice ends with a letter, the rule likely involves letters).
7	Time‑saving trick	If you’re stuck, test a simple rule (e.g., “add 3”) on all given pairs; if it fails, move to the next simplest rule.

3. Worked Example

Question

Code	Decoded
125	12
379	16
246	20

Which of the following is the correct decoding for 431?

Option	Decoded
A	13
B	19
C	21
D	22

Step‑by‑Step Solution

Step	Action	Result
1	Look at the numbers – They are 3‑digit codes.
2	Test a simple rule – Maybe “sum of the digits”?
3	Sum digits of 125 → 1+2+5 = 8 (not 12). So not sum.
4	Try “difference between first and last digit”? 5‑1 = 4 (not 12).
5	Notice that the decoded number is close to the sum of the two outer digits: 1+5 = 6; 6+? = 12 → add 6.
6	Check 379: 3+9 = 12; 12+? = 16 → add 4.
7	The added value seems to be the middle digit (2→? no).
8	Instead, try “(first digit × last digit) + middle digit”? 1×5 = 5; 5+2 = 7 (not 12).
9	Re‑evaluate: maybe the rule is “(first digit × last digit) + (second digit × 2)”? 1×5=5; 2×2=4; 5+4=9 (no).
10	Pattern emerges: 12 = 1+2+5 + 4; 16 = 3+7+9 + 1; 20 = 2+4+6 + 8. The added value seems to be half of the sum of digits? 1+2+5=8; half=4 → 8+4=12. 3+7+9=19; half≈9.5 (not integer).
11	Try “sum of digits + product of digits”: 1+2+5=8; 1×2×5=10; 8+10=18 (not 12).
12	Look at the decoded numbers: 12, 16, 20 – they are all even and increase by 4 each time. Maybe the decoding rule is “sum of digits × 2”? 1+2+5=8; 8×? = 12 → ×1.5 (not integer).
13	New idea – Maybe the rule is “(first digit + last digit) × middle digit”? 1+5=6; 6×2=12 ✔️. 3+9=12; 12×?=16 → 12×1.333… (no).
14	Check another rule – “(first digit + last digit) × (middle digit ÷ something)”.
15	Instead of inventing, test simple arithmetic on the whole code: 125 ÷ 12 ≈ 10.416; 379 ÷ 16 ≈ 23.687; 246 ÷ 20 = 12.3. No pattern.
16	Try mapping to letters: 125 → 12 → L? 379 → 16 → P? 246 → 20 → T? The decoded numbers 12, 16, 20 correspond to letters L, P, T. Are these letters related? They are 4 letters apart. 12→L, 16→P, 20→T (difference +4). So maybe the decoded number is the alphabet position of a letter that is 4 letters ahead of the first digit? 1→E (5), 3→G (7), 2→F (6) – no.
17	Switch approach – maybe the rule is “(sum of first two digits) + (sum of last two digits) / 2”? 1+