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The symbols $\triangle$, $\square$, $\diamond$, $\clubsuit$ represent four different integers from 1 to 9. Using the equations below, what is the value of $\square$?

$\triangle + \triangle + \triangle + \triangle + \triangle = \diamond$
$\triangle + \square + \square = \diamond$
$\square + \diamond = \clubsuit$

 Feb 25, 2021
 #1
avatar+964 
+2

triangle = T, square = S, diamond = D, club = C

 

5T = D                . (1)

T + 2S = D         . (2)

S + D = C            . (3)

 

Equations (1) and (2) are equal:

 

5T = T + 2S

4T = 2S

2T = S

 

Then we can look at equation (3):

 

S + T + 2S = C

7S/2 = C

 

Every symbol is an integer from 1 to 9, so S = 7.

 Feb 25, 2021
 #2
avatar+394 
+3

T=Triangle, D=Diamond, S=Square, and C=Clover

 

T+T+T+T+T=D 

T can't be larger than 1 because D would be larger than 9.

 

So this means 1+1+1+1+1=5, which D=5

 

1+S+S=5

1+2S=5

2S=4

S=2

 

Square is equal to 2

 

S+D=C = 2+5=7, so C is equal to 7

 

T=1, D=5, S=2, C=7

 

I know CubeyThePenguin got a different answer, so I'm not sure if Cubey's or mine is correct :)

 Feb 25, 2021

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