Can a sum name its pair?
Choose ten different numbers from 1 to 90. Make every pair sum different.
The clock starts when you reveal the puzzle.FORTY-FIVE LITTLE FINGERPRINTS
Two cards.
One unique sum.
Ten numbers.
No pair can hide.
Use two different cards per pair. The order of the cards does not matter.AT YOUR OWN PACE
A brain snack, with room to think.
THE ANSWER OPENS ANOTHER DOOR
Now for the lovely part.
You have an answer. Explore how it works, then take the idea somewhere new.
See why it works ↓Keep your time
EXPLORE YOUR TEN NUMBERS
A sum becomes a fingerprint.
Pick any sum below. Exactly one pair in your own set produces it.
ONE WAY TO BUILD A SET
Give the next number some space.
Suppose every existing pair has a unique sum. Add a new number at least as large as the biggest existing pair sum. Because all cards are positive, every pair using the new number now exceeds every old pair sum. And those new sums differ from one another because the old cards differ.
1 · 2 · 3 · 5 · 8 · 13 · 21 · 34 · 55 · 89
After the first two cards, each new number here is the sum of the two largest previous numbers. The construction keeps the old and new sums apart. Your own set may use a different pattern.
Ten cards make 10 × 9 ÷ 2 = 45 pairs: pick a first card, then a different second card, then divide by two because swapping their order does not make a new pair.
The idea to take away
A unique output can identify its inputs. Designing those fingerprints means watching for collisions, rather than merely making the numbers different.