# 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.4 Formulaic

Intuitive Leap: The symbols stand for numbers in a signed-digit base 5 number system.

The lines are:

```T  þ  þ/Ð  þ\þ  ÐXØ    þ/ÐXØ  þ/þ\þ  þ\þ/Ð  þ\Ð\þ    Ð/þXØ    Ð\þXØ    þ/Ð/Ð\þ  þ/Ð\þ/Ð
S  þ  þ/þ  Ð/þ  þ/Ð\þ  þXØXØ  þ\Ð\þ  ÐXØ/þ  þ/Ð/Ð/þ  þ/Ð\þ\þ  þ/þXØXØ  þXØ/þ\þ  þ\þ/þ/þ
P  Ð  þ/Ð  þXØ  þ\Ð    Ð\þ    þ/Ð/Ð  þ/Ð\Ð  þ/þ/þ    þXØ/Ð    þ\þ/þ    þ\þ\þ    þ\Ð\Ð
F  þ  þ    Ð    þ/Ð    þXØ    Ð/Ð    þ/Ð/Ð  þ/þ\þ    þ\Ð/þ    Ð\þXØ    þ/þ/Ð/þ  þ\þ/þ/þ
```

Each line is a numerical sequence which can be identified by looking for patterns in the symbol groups. There are several different symbols being used:

```"þ", "Ð", "/", "\", "X", and "Ø".
```

How do these symbols represent numbers?

There are many methods that could be used to identify what the sequences are. One way to start is to notice that the bottom sequence (marked "F") begins with two repeating symbols, and to remember that the Fibonacci sequence, well known to all mathematicians, has this property. It's formed by adding the previous two digits together to make a new digit. The sequence goes:

```F  1  1    2    3      5      8      13     21       34       55       89       144
```

If this is correct, we have the following mappings between symbols and numbers:

```1 þ
2 Ð
3 þ/Ð
5 þXØ
8 Ð/Ð
13 þ/Ð/Ð
21 þ/þ\þ
34 þ\Ð/þ
55 Ð\þXØ
89 þ/þ/Ð/þ
144 þ\þ/þ/þ
```

Assuming all rows use the same number system, we can replace symbols in the other sequences with numbers we've discovered from the Fibonacci row:

```T  1  3    þ\þ  ÐXØ    þ/ÐXØ  21     þ\þ/Ð  þ\Ð\þ    Ð/þXØ    55       þ/Ð/Ð\þ  þ/Ð\þ/Ð
S  1  þ/þ  Ð/þ  þ/Ð\þ  þXØXØ  þ\Ð\þ  ÐXØ/þ  þ/Ð/Ð/þ  þ/Ð\þ\þ  þ/þXØXØ  þXØ/þ\þ  144
P  2  3    5    þ\Ð    Ð\þ    13     þ/Ð\Ð  þ/þ/þ    þXØ/Ð    þ\þ/þ    þ\þ\þ    þ\Ð\Ð
```

The P row now holds enough numbers to identify it as the first prime numbers:

```P  2  3    5    7      11     13     17    19        23      29        31       37
```

Using the numbers found so far, the top two rows can be identified as the Triangular numbers and the Square numbers respectively:

```T  1  3    6    10     15     21     28     36       45       55       66       78
S  1  4    9    16     25     36     49     64       81       100      121      144
```

This gives us a lot more number mappings, and we can hopefully decode the numbering scheme now:

```  1 þ     11 Ð\þ    21 þ/þ\þ  31 þ\þ\þ  41
2 Ð     12        22        32        42
3 þ/Ð   13 þ/Ð/Ð  23 þXØ/Ð  33        43
4 þ/þ   14        24        34 þ\Ð/þ  44
5 þXØ   15 þ/ÐXØ  25 þXØXØ  35        45 Ð/þXØ
6 þ\þ   16 þ/Ð\þ  26        36 þ\Ð\þ  46
7 þ\Ð   17 þ/Ð\Ð  27        37 þ\Ð\Ð  47
8 Ð/Ð   18        28 þ\þ/Ð  38        48
9 Ð/þ   19 þ/þ/þ  29 þ\þ/þ  39        49
10 ÐXØ   20        30        40        50

55 Ð\þXØ
64 þ/Ð/Ð/þ
66 þ/Ð/Ð\þ
78 þ/Ð\þ/Ð
81 þ/Ð\þ\þ
89 þ/þ/Ð/þ
100 þ/þXØXØ
121 þXØ/þ\þ
144 þ\þ/þ/þ
```

There are some interesting patterns emerging:

• All multiples of five end in XØ.
• A repeating pattern of \ \ / / X appears in groups of five increasing numbers.
• A repeating pattern of þ Ð Ð þ Ø appears as the final symbols in groups of five increasing numbers.

All of these patterns involve 5 - and it is in fact a base 5 number system. But it is not ordinary base 5. It's a signed-digit number system, so instead of the digits 0 to 4 for each numeral, the digits -2 to +2 are used.

• "/" means -.
• "\" means +.
• "þ", "Ð", and "Ø", mean 1, 2, and 0 respectively.
• "X" means "+" or "-" - it doesn't matter because it's always followed by zero.

Here are some examples of how numbers are constructed in this number system:

```         25's   5's  1's
1=þ                  1
2=Ð                  2
3=þ/Ð           1   -2
4=þ/þ           1   -1
5=þXØ           1   +0
6=þ\þ           1   +1
7=þ\Ð           1   +2
8=Ð/Ð           2   -2
9=Ð/þ           2   -1
10=ÐXØ           2    0
21=þ/þ\þ   1    -1   +1
55=Ð\þXØ   2    +1    0
```

Now we can decode the numbers in the second part of the puzzle:

```þ/ÐXØ/Ð\ÐXØ
þ/ÐXØ\þ\ÐXØ
þ/Ð\þ/þ\Ð\þ
þ/Ð\þ\Ð\Ð\þ
```

They are all 6-digit numbers (in our signed-digit base-5 number system anyway):

```5  4  3  2  1  0    (Powers of 5: 3125, 625, 125, 25, 5, 1)
1 -2  0 -2 +2  0  = 1x3125 + -2x625 + 0x125 +-2x25 + 2x5 + 0x1 = 1835
1 -2  0 +1 +2  0  = 1x3125 + -2x625 + 0x125 + 1x25 + 2x5 + 0x1 = 1910
1 -2 +1 -1 +2 +1  = 1x3125 + -2x625 + 1x125 +-1x25 + 2x5 + 1x1 = 1986
1 -2 +1 +2 +2 +1  = 1x3125 + -2x625 + 1x125 + 2x25 + 2x5 + 1x1 = 2061
```

The numbers 1835, 1910, 1986, and 2061 look remarkably like years! And they are spaced about 75 or 76 years apart. A good knowledge of astronomy, or a search on the Internet reveals that they are years when Halley's Comet makes a close approach to Earth and the Sun. And indeed the answer is HALLEYS COMET.