# Math Puzzle for Quarantining Humans: 3333

My next few posts will be a series of simple math puzzles that will require a little bit of cleverness to solve. Here’s the list if you want to jump to a particular problem:

- 3333 (this post)
- 6 Inscribed Circles
- Slicing the Parabola
- 4444
- […coming soon…]
- […coming soon…]

Hopefully, some bored human in quarantine might find joy in solving these problems. Let’s start with `3333`

:

# 3333: PROBLEM

You have one number minus another number, and the result is 3333. The question is: what are the 2 numbers that give this result? But here’s the catch: the digits of the two numbers must be chosen from 1–9, and each digit must be used at exactly once.

For instance, `5678 — 2345`

does not work because it does not use the digit `1`

and it uses the digit `5`

twice. `12345 — 6789`

is invalid because the result is not `3333`

.

Now, I already gave you a hint in the picture above that the first number must have 5 digits and the second number 4 digits. That’s the only way this would work. Now, you have to find the two numbers. GO!

**Scroll down to see a hint if you get stuck, and scroll further down to see the solution.**

# 3333: HINT

There’s a high possibility that you figured this out already, but here it goes. Since the operation’s result (one number minus another) is 3333, a positive number, the top number must be larger than the bottom number, and since we need to use all 9 digits, the first digit of the top number must be a small one (because if it’s too large then the result will be too large). In fact, like mentioned above, it’s 5 digits and the bottom is 4 digits. Thus, the first digit must be a 1. It can’t be 2 because 20,000–9,999 = 10,001 (the smallest difference between a 5 digit number starting with 2 and a 4 digit number).

# 3333: SOLUTION

It turns out that there are many solutions to this problem, and here they are:

`12678 - 9345 = 3333`

12687 - 9354 = 3333

12768 - 9435 = 3333

12786 - 9453 = 3333

12867 - 9534 = 3333

12876 - 9543 = 3333

If you look closely, the solutions resemble each other a lot, and they reveal something about how you find the solution.

First, we know we’re doing something of the form `1xxxx — xxxx = 33333`

. Because we know that 1 is the first digit (as explained in the hint), we know that we must carry in order for that 1 to disappear. Let’s keep that information aside for now.

We need to list all the possible ways we get get `3`

from subtracting two digits. We get it with `4-1`

, `5-2`

, `6-3`

, `7-4`

, `8-5`

, `9-6`

. We can’t use `4-1`

because the digit `1`

is already used. At the same times, we can use at most 3 of these because after 3 selections of pairs of digits, other digits start repeating (this is because any group of more than 3 pairs will repeat a digit).

It seems like we’re out of luck. However, remember that we have to carry over to get rid of the `1`

. Which means, there are other possibilities at play. If we carry, we can get `10`

plus whatever digit was there before. All the ways we can get `3`

from `10`

plus a digit minus another digit are: `10+1-8`

and `10+2-9`

. Those give us `1-8`

and `2-9`

pairs. However, `1`

again cannot be used, so we’re left with only one pair: `2–9`

. However, that’s just enough to get one of the solutions. We can use the following pairs: `2-9`

, `8–5`

, `7-4`

, `6-3`

. With that, we get `12876 — 9543 = 3333`

.

Now, you can see that the order of the last 3 pairs that I gave has nothing inherently special. That is, the last 3 pairs could be permuted. The permutation formula is the following:

What this formula essentially describes is the following. Let’s say we have `n`

objects, and let’s say we could order these objects into `r`

possible placements. Then the total number of orderings is given by `P(n,r)`

. Here’s a video that probably explains it better. With that said, `P(3,3) = 3!/(3-3)! = 3!/0! = 6`

. If you look above, we have exactly 6 solutions! So, that checks out with the solutions we gave above :D.

**If you were able to solve this, why don’t you try ****4444****? It’s the same problem, except the result is 4444. It’s much harder than this one, so get ready!**