# CISRA Puzzle Competition 2007 - Solutions

This is the archive of the 2007 Puzzle Competition. Please visit the current competition site for information about the latest Puzzle Competition.

## C.2 Calculate

Intuitive Leap: "+" and "-" refer to adding or subtracting segments of the digits on a calculator 7-segment display.

Here's the lesson:

```0 = (9 - 4) + (8 - 3) + 1
1 = (8 - 2) - (9 - 3) + (9 - 5)
2 = (3 - 1) + (8 - 5)
3 = (9 - 4) + (8 - 0) + 1
4 = (9 - 3) + (8 = 0) + 1
5 = (3 - 1) + (8 - 2)
6 = 8 - (9 - 5)
7 = A - (6 - 5) - (8 - 0)
8 = 9 + (6 - 5)
9 = (8 - 2) + (8 - 6) + (3 - 1)
```

We can re-write the first equation using the 7 segment display of a cheap calculator like so:

``` _       _               _    _
| | = ( |_| - |_| ) + ( |_| - _| ) +   |
|_|      _|     |       |_|   _|       |
```

Taking the segments of "4" away from the segments of "9" leaves two segments, on the top and the bottom. Likewise, "8" - "3" leaves two vertical segments:

``` _       _
| | =               +   |          +   |
|_|      _              |              |
```

Putting all of the segments together on the right hand side does in fact give "0".

The other equations can be seen to hold too. The puzzle text draws special attention to the number 7. Let's look at it:

``` _     _       _     _         _     _
| = |_| - ( |_  - |_  ) - ( |_| - | | )
|   | |     |_|    _|       |_|   |_|
```

which becomes

``` _     _
| = |_| -               -    _
|   | |     |
```

which becomes

``` _     _
| = | |
|     |
```

which tells us that the digit '7' must have a hanging descender on the top-left of its representation.

The homework asks for solutions to several equations:

```? = 8 - (9 - 4)
? = 7 + (6 - 5) + (8 - 0)
? = (6 - 5) + (8 - 0)
? = (8 - 7) + 1
? = 0 - 1
? = (9 - 5) + (8 - 4) + (8 - 2)
? = 2 - (9 - 4) - (8 - 6)
? = 8 - 1
```

We can now "calculate" the answers to each line:

```H = 8 - (9 - 4)
A = 7 + (6 - 5) + (8 - 0)
r = (6 - 5) + (8 - 0)
d = (8 - 7) + 1
C = 0 - 1
O = (9 - 5) + (8 - 4) + (8 - 2)
r = 2 - (9 - 4) - (8 - 6)
E = 8 - 1
```

The answer is revealed as HARDCORE.