Can three totals be squares?
Place 1 to 9 once each. The four numbers in every larger triangle must add to a perfect square.
The clock starts when you reveal the puzzle.THREE TOTALS. NINE SHARED POSSIBILITIES.
Think in triangles.
Make squares.
Nine numbers.
Three overlapping totals.
Select a triangle, then a number. Numbers already placed swap positions.
Tap a total to highlight its whole region. Keyboard: select a cell, then type 1–9.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
INSPECT YOUR THREE TOTALS
Some numbers work twice.
Explore each outlined region in your own arrangement. Shared cells connect the three conditions.
Each region uses four distinct numbers. A square total in one region is only part of the challenge.
NARROW THE POSSIBILITIES
Only two squares can fit.
The smallest four-number total is 1 + 2 + 3 + 4 = 10. The largest is 6 + 7 + 8 + 9 = 30. Only 16 and 25 are perfect squares between them.
Then count the overlap.
The nine numbers add to 45. Adding all three region totals counts every number once, and the three shared numbers a second time.
The three shared numbers total between 6 and 24, so the region totals together must lie between 51 and 69. Three 16s give only 48; three 25s give 75. Both are impossible. Two 16s and one 25 give 57, leaving a shared total of 12. One 16 and two 25s give 66, leaving 21.
That restriction helps you search. It does not, by itself, prove that all three regions will work: your completed arrangement does that.
The idea to take away
When several conditions overlap, look at the shared parts. Adding the conditions can expose a hidden constraint.