Lens 01 of 9

The short list

Look at the condition that leaves only a handful of candidates. List them, then check the rest.

Some conditions don't narrow the answer, they almost name it. 'The product of the digits is 14' allows exactly one pair of digits. When a condition does that, write the whole list out and test the other conditions on it. No equation is needed, because the list is the equation, already solved.

The clues that call for it

Quadratic Equations

The product of the digits is given

Product 14 → only 2 × 7. The number is 27 or 72.

Quadratic Equations

Two numbers with a given product

182 = 2 × 7 × 13. Group the primes: 13 and 14 sum to 27.

Quadratic Equations

A fraction whose numerator and denominator are tied together

Numerator + denominator = 10 → 1/9, 3/7, 7/3, 9/1.

Anywhere

A small whole-number answer with a tight condition

Sum 8 → (1, 7), (2, 6), (3, 5), (4, 4).

Worked example

A two-digit number has digit product 14. Add 45 and its digits reverse. Find it.

1
Take the tightest condition first
The digit product is 14, and digits live in 1 to 9.
d₁ × d₂ = 14
2
List the whole space
Only 2 × 7 makes 14, so there are exactly two candidates.
- 27
- 72
3
Filter with the leftover condition
Add 45 and check whether the digits really reverse.
27 + 45 = 72 ✓ · 72 + 45 = 117 ✗
- 27
- 72
4
Answer
27. No equation was written: the condition collapsed the space to two numbers.

Try the lens

The short list

The clues that call for it

A two-digit number has digit product 8, and adding 18 reverses its digits. Which is it?

Problems that use this lens