If wxyz is a square, which statements must be true?
Ever stare at a four‑digit number like 6249 and wonder if it could be a perfect square? Maybe you’ve seen puzzles that ask, “Find all four‑digit squares whose digits are w, x, y, z.” The moment you realize the answer hinges on a handful of digit‑level rules, the puzzle clicks.
Below is the full rundown of the must‑be‑true statements when the four‑digit number wxyz is a perfect square. I’ll walk through the logic, flag the common traps, and give you a cheat‑sheet you can actually use the next time a brain‑teaser pops up Most people skip this — try not to. Still holds up..
Short version: it depends. Long version — keep reading.
What Is wxyz Being a Square?
When we say wxyz is a square, we mean the integer formed by the digits w, x, y, z (in that order) equals n² for some integer n. Basically,
wxyz = n² where 32 ≤ n ≤ 99
Why 32 to 99? Because 31² = 961 (only three digits) and 100² = 10 000 (five digits). So any four‑digit square lives somewhere in that narrow band.
Think of wxyz as a string of four separate digits, each ranging from 0 to 9, with w ≠ 0 (otherwise it wouldn’t be a four‑digit number). The job is to translate the “square” condition into statements about those individual digits No workaround needed..
Why It Matters
Understanding the digit constraints saves you hours of brute‑force checking. Instead of testing every number from 1024 to 9801, you can eliminate most candidates instantly Simple, but easy to overlook. Practical, not theoretical..
In practice, this is the secret sauce behind many math‑competition problems and logic puzzles. It also shows up in coding challenges where you need to filter a list of numbers quickly. Knowing the “must‑be‑true” statements lets you write a tiny piece of code or a mental shortcut that cuts the search space by 80 % or more.
How It Works
Below are the concrete statements that must hold whenever wxyz is a perfect square. I’ll break them into bite‑size chunks, give a quick proof, and note any exceptions.
1. The Last Digit (z) Is One of 0, 1, 4, 5, 6, 9
A square can only end in certain digits. The reason is simple: look at the units digit of any integer and square it.
| n (units) | n² (units) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 6 |
| 5 | 5 |
| 6 | 6 |
| 7 | 9 |
| 8 | 4 |
| 9 | 1 |
So z ∈ {0, 1,
