Letter land
Problem
If
$A + C = A$
$F \times D = F$
$B - G = G$
$A + H = E$
$B / H = G$
$E - G = F$
and $A$-$H$ represent the numbers from $0$ to $7$
Find the values of $A, B, C, D, E, F, G$ and $H$.
Getting Started
$C$ and $D$ first
Student Solutions
We can simplify most of these equations first, before we start giving values to all of the letters.
As $A + C = A$ we know that $C = 0$.
As $F \times D = F$ we know that $D = 1$.
$B - G = G$, therefore $B = 2G$ (add $G$ to both sides of the
equation).
$B / H = G$ therefore $B = HG$ (multiply both sides by $H$).
Since $B = HG$, and $B$ also equals $2G$,
$H$ must be $2$.
Since $B$ is twice $G$, the options are:
| $B$ | $G$ | Can this work? |
| $2$ | $1$ | No, as $H+2$ and $D=1$ already. |
| $4$ | $2$ | No, as $H=2$ already. |
| $6$ | $3$ | Yes, this is possible. |
So $B = 6$ and $G = 3$
$E - G = F$, therefore $E - 3 = F$.
The smallest possible value for $F$ is $4$, and this means that $E$
would be $7$.
$F$ cannot be any greater than $4$ because this would mean that $E
> 7 $ and this is not allowed.
Therefore $F = 4$ and $E = 7$
Now we can solve $A + H = E$.
This is $A + 2 = 7$, so $A = 5$.
Here are the solved equations:
| $A + C = A$ | $5 + 0 = 5$ |
| $F \times D = F$ | $4 \times1 = 4$ |
| $B - G = G$ | $6 - 3 = 3$ |
| $A + H = E$ | $5 + 2 = 7$ |
| $B / H = G$ | $6 / 2 = 3$ |
| $E - G = H$ | $7 - 3 = 4$ |
And all the values of $A$-$H$:
| $A$ | $5$ |
| $B$ | $6$ |
| $C$ | $0$ |
| $D$ | $1$ |
| $E$ | $7$ |
| $F$ | $4$ |
| $G$ | $3$ |
| $H$ | $2$ |