# Fruity system of equations

I've seen a lot of these popping up in Facebook lately so I decided to make my own. Can you tell me what each symbol is?

There are no math errors in this puzzle.

In order for your solution to be correct it must work in all 4 equations.

Hint #1

As pointed out by Silenus below, changing the equals sign would allow for many trivial solutions. So I will confirm that the equals sign represents equality. and also, numbers represent their actual numerical value.

Hint #2

The only numbers that show up in the solution are positive integers.

Hint #3

One of the fruit is not like the others. It represents a symbol on your keyboard which is neither a number nor a letter.

Hint #4

Each symbol maps to a single character which could be found on an ordinary keyboard. This means that no numbers in the solution are larger than 9.

Hint #5

The third equation has the string apple carrot banana, but if I reordered this expression to carrot apple banana, it would be nonsensical (it's a syntax error, something your calculator would not let you compute).

• Is star multiplication or something else? Dec 18, 2016 at 0:17
• @wildBillMunson Each symbol behaves the same way every time it shows up. To solve this one you'll have to think outside the fruit basket. Dec 18, 2016 at 0:38
• Can we assume all numbers are equal to their usual representation and that the equals sign denotes equivalence? Dec 18, 2016 at 1:14
• Guys try interpreting the + and * as variables too.. Dec 18, 2016 at 1:18
• I've been trying to solve this with the fruits not representing a numerical value, but instead an action. For instance, letting apple = +3 and banana = +3 and carrot = +18, we get a remarkably close answer of everything working out apart from the last line equalling 78. Dec 18, 2016 at 2:14

The symbols are as follows:

$$\begin{array}{rl}\bf{Apple:}&3\\\bf{Banana:}&1\\\bf{Carrot:}&\ ! \ \ \ \ (factorial)\\+:&+\\*:&4\end{array}$$

That gives:

$\ \ \ \ 3+3+3=9$
$\ \ \ \ 3·1·4·1·3=36$
$\ \ \ \ 3!·1·4·3!=144$
$\ \ \ \ 3·4!+3!·3=90$

all of which are true.
note: $\ ·\$ denotes multiplication due to adjacency, per OP comments.

Banana can also be $\times$ and still fulfill all the equations. Only a strict reading of Hint 3 prevents it from being a solution (and I hadn't noticed that hint, so my original answer here used $\times$):

$\ \ \ \ 3+3+3=9$
$\ \ \ \ 3\times4\times3=36$
$\ \ \ \ 3! \times 4·3!=144$
$\ \ \ \ 3·4!+3!·3=90$

are all equally true as well.

• Awesome, you got it!!! Side note: I originally make banana *, but realized later it could be 1 as well. Dec 20, 2016 at 16:58

From hints and comments we know that:

• juxtaposition of variables or numbers means multiplication;
• each symbol represents a symbol on the keyboard so that valid number replacements go from 0 to 9;
• one symbol isn't a letter or digit;
• the equals sign and the numbers on the right hand side represent themselves.

Further, let's assume that:

• the plus sign really is the plus sign;
• the asterisk does not mean multiplication, since that is already signalled by juxtaposition and
• the asterisk represents a symbol or operator.

The replacements should then be:

apple: 3
banana: 1
asterisk: 4
carrot: ! (exclamation mark)

The exclamation marks stands for factorial when put after an expression. That's why no expression can start with the carrot. The final equations are:

9 = 3 + 3 + 3
36 = 3·1·4·1·3
144 = 3!·1·4·3! = 6·1·4·6
90 = 3·4! + 6·3 = 3·24 + 6·3

I've looked at this multiple ways and it appears to me that this puzzle's solution is:

Non-existent

I determined this by solving each equation one by one.

The first equation can only be

3+3+3=9 as it's the same number the times

Thus the makes the second equation

3b*b3=36
So b =2

Here's where thing get complicated. If you solve the 144 equation first, you get

Carrot =2.8... Give or take

But if you solve the 90 equation first, you get

Carrot = 7.5

Here's a picture of my work (sorry if it's oversized I'm uploading this on mobile)

• Lateral thinking tag... Dec 18, 2016 at 0:55
• Didn't see it :) Dec 18, 2016 at 0:56

I tried many things, even lateral thinking and this equation simply cannot work, even by changing a fruit for (X,+,-,...) or even by changing the * symbol for another one.

The only way it works that I found is.

Apple : 100
Banana : 101
Carrot : 102
"+ symbol" : +
"* symbol" : *
"= symbol" : >
The question did say to find the value of every symbols, not every fruit so... might be good.

• If '=' means '>', the equations have infinite solutions for large values of the fruit, which is probably incorrect. I would think "find the value of every symbol" includes at least the fruit symbols but possibly also the mathematical operation symbols. Dec 18, 2016 at 1:40
• @Silenus I would agree with your first thought, but as for the second thought, I would go more with definitely includes the math symbols, seeing as it says "every symbol" Dec 18, 2016 at 2:26
• However, they did technically find a solution for each Dec 18, 2016 at 3:48

Partial -

First equation is ->

Apple + Apple + Apple = 9

Hence -

If Apple = 3 we get 3+3+3 = 9
Apple = 3

Second Equation is ->

Apple Banana * Banana Apple

Hence - >

If Banana = * and * = 4 we get 3*4*3 = 36
Banana = * and * = 4