Problems & Puzzles: Puzzles

 Puzzle 611. Puzzle 2-3-5-7. Stephen Johnson and me constructed the following puzzle, which is a variation of some others constructed before out there of the same nature: C1: Using at the most four of these decimal digits: 2, 3, 5 & 7 where repetitions are allowed & counted. C2: Using any of these arithmetical six symbols: +, -, *, ^, ( & ) where repetitions are allowed & counted. Q. Get all the numbers from 0 to X in its minimal expression (minimal quantity Q of digits & arithmetical symbols) and stop whenever you can not get X+1 following the conditions C1 & C2. Valid examples: Follow the rules C1 & C2 and Q= minimal. 21=3*7, Q=3 13=3+3+5, Q=5 22=22, Q=2 24=3*7+3, Q=5 Invalid examples: Q is not minimal &/or fail at using C1 &/or C2. 13=22-3*3, Q=6, (Not minimal) 5=2+2+2/2, Q=7, (Not minimal & using an invalid arithmetical symbol) 24=2+3+5+7+7, Q=9, (Not minimal & using more than four prime integer) 4=4 Q=1, (Using an invalid integer)

Contributions came from Giovanni Resta, Claudio Meller, Hakan Summakoğlu.

***

All of them discovered that the first integer that can not be produced according to the rules given, is 179 and sent their respective expressions for the integers from 0 to 178.

It happened that the total sum of the symbols used in these integers from 0 to 178 is 774.

In their respective lists sent by the first time, Giovanni & Hakan used 775 symbols, while Claudio used 805 & Carlos Rivera used 809 symbols.

There are equivalent expressions for some integers that use the same quantity of symbols. For example: 20 = 23-3 = 22-2

This is one list using 174 symbols

 n Q Expression 0 3 2-2 1 3 3-2 2 1 2 3 1 3 4 3 2+2 5 1 5 6 3 3+3 7 1 7 8 3 2^3 9 3 3^2 10 3 2*5 11 5 2^3+3 12 3 5+7 13 5 35-22 14 3 7+7 15 3 3*5 16 4 23-7 17 4 22-5 18 4 23-5 19 4 22-3 20 4 23-3 21 3 3*7 22 2 22 23 2 23 24 4 2+22 25 2 25 26 4 33-7 27 2 27 28 4 33-5 29 4 22+7 30 4 32-3 31 4 33-2 32 2 32 33 2 33 34 4 32+2 35 2 35 36 4 33+3 37 2 37 38 4 33+5 39 4 32+7 40 4 33+7 41 5 73-32 42 4 37+5 43 5 75-32 44 4 2*22 45 4 52-7 46 4 2*23 47 4 52-5 48 4 55-7 49 3 7*7 50 4 2*25 51 4 53-2 52 2 52 53 2 53 54 4 2*27 55 2 55 56 4 3+53 57 2 57 58 4 3+55 59 4 2+57 60 4 5+55 61 6 22*3-5 62 4 5+57 63 5 3*3*7 64 4 2*32 65 4 72-5 66 4 2*33 67 4 72-5 68 4 73-5 69 4 23*3 70 4 2*35 71 4 73-2 72 2 72 73 2 73 74 4 2*37 75 2 75 76 4 73+3 77 2 77 78 4 73+5 79 4 77+2 80 4 77+3 81 4 3*27 82 4 77+5 83 6 3+7+73 84 4 7+77 85 5 33+52 86 5 33+53 87 5 55+32 88 5 33+55 89 5 57+32 90 5 35+55 91 6 2*7+77 92 5 35+57 93 6 2^7-35 94 5 22+72 95 5 22+73 96 4 3*32 97 5 25+72 98 5 2*7*7 99 4 3*33 100 5 25+75 101 6 2^7-27 102 5 27+75 103 6 3*32+7 104 4 2*52 105 4 3*35 106 4 2*53 107 5 55+52 108 5 55+53 109 5 57+52 110 4 2*55 111 4 3*37 112 5 55+57 113 6 2+3*37 114 4 2*57 115 4 23*5 116 6 3*37+5 117 6 2*55+7 118 5 5^3-7 119 6 5+2*57 120 5 5^3-5 121 5 2^7-7 122 5 5^3-3 123 5 2^7-5 124 5 72+52 125 3 5^3 126 5 2^7-2 127 5 5^3+2 128 3 2^7 129 5 77+52 130 5 5^3+5 131 5 2^7+3 132 5 5^3+7 133 5 2^7+5 134 5 77+67 135 4 5*27 136 7 2^7+2^3 137 6 2+27*5 138 6 2*3*23 139 6 2*73-7 140 6 2*2*35 141 6 2*72-3 142 6 27*5+7 143 6 2*73-3 144 4 2*72 145 5 72+73 146 4 2*73 147 5 3*7^2 148 5 73+75 149 5 77+72 150 4 2*75 151 6 2^7+23 152 5 75+77 153 6 2^7+25 154 4 2*77 155 6 5+2*75 156 4 3*52 157 6 32+5^3 158 6 33+5^3 159 4 3*53 160 4 32*5 161 4 7*23 162 6 2+32*5 163 6 2^7+35 164 6 5+3*53 165 4 55*3 166 6 3^5-77 167 6 2+33*5 168 6 3+33*5 169 6 3*57-2 170 6 3^5-73 171 4 3*57 172 6 7+5*33 173 6 2+3*57 174 6 3*57+3 175 4 5*35 176 6 5+3*57 177 6 5^3+52 178 6 3+35*5

***

 Records   |  Conjectures  |  Problems  |  Puzzles