CISRA Puzzle Competition 2007  SolutionsThis is the archive of the 2007 Puzzle Competition. Please visit the current competition site for information about the latest Puzzle Competition. C.4 FormulaicIntuitive Leap: The symbols stand for numbers in a signeddigit 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 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 signeddigit number system, so instead of the digits 0 to 4 for each numeral, the digits 2 to +2 are used.
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 6digit numbers (in our signeddigit base5 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.
