Title here
Summary here
A Christmas Binary Miracle
December 26, 2013 in Systems9 minutes
My brother got a little puzzle in his stocking this Christmas. It was a little cardboard booklet, and on each page was written a block of numbers, like so:
BLOCK ONE
1 3 5 7 9 11 13 15
17 19 21 23 25 27 29 31
33 35 37 39 41 43 45 47
49 51 53 55 57 59 61 63
BLOCK TWO
2 3 6 7 10 11 14 15
18 19 22 23 26 27 30 31
34 35 38 39 42 43 46 47
50 51 54 55 58 59 62 63
BLOCK THREE
4 5 6 7 12 13 14 15
20 21 22 23 28 29 30 31
36 37 38 39 44 45 46 47
52 53 54 55 60 61 62 63
BLOCK FOUR
8 9 10 11 12 13 14 15
24 25 26 27 28 29 30 31
40 41 42 43 44 45 46 47
56 57 58 59 60 61 62 63
BLOCK FIVE
16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 31
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63
BLOCK SIX
32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63You’re to ask someone to pick any number they see in any block, and don’t tell you what it is. They are, however, required to find every instance of that number in the entire booklet, and tell you which blocks of numbers that specific number shows up in. Of those blocks that they’ve identified, you as the “puzzler” are supposed to add the first number (top left corner) in each of those blocks, and the resulting number will be the number they selected.
Try it out. Let’s say our number is 54. The number 54 is present on blocks 2, 3, 5, and 6. The first numbers in those blocks are 2, 4, 16, and 32 respectively. The sum of those four numbers is 54.
Pretty cool, eh?
Spoilers Below!!
I don’t post a lot of purely math-related content, but I did so here because there’s a parallel between the math behind this puzzle and network engineering.
If you take a look at the first number in each block, you’ll notice they’re all significant decimal numbers when translating to binary. These are all the highest values that each binary digit, in order, can represent.
For instance, the first bit in a binary string can only describe, at most, the number 1. A second bit is twice that, because a binary string of “10” is equal to 2, and so on up to the 6th block, which is equivalent to a 6-bit string (100000 in binary equals 32). So, if you assign bit values to each block of numbers, it correlates to a binary digit.
Let’s pick on our example again. We had the number 54, and noticed that it was present in blocks 2, 3, 5, and 6. Well, if we take our six blocks, and represent each with a binary digit - setting that digit to 1 if the number is present, and 0 if it is not, we get “011011”. That is the binary equivalent to the number 54.
It follows, then, that each block contains literally all numbers that have that respective bit value set to 1 in a six-bit binary number. Take block 3 for example. The third bit (that holds the “4” value in decimal) is set to 1 for every single number in that block.
What you’re doing by solving this puzzle is actually what network engineers do when they do binary conversions, such as when subnetting. It binary, or base2 math at it’s core, and functions exactly the same way.
Now that’s cool.
Glutton for Punishment
Because of the parallels with network engineering, you may also think of a 6-bit puzzle as a little…….incomplete. We like to deal with binary values in 8-bit chunks. So I set out to see the impact of adding two bits to this puzzle. After all, how hard could it be, just adding two bits? Keep in mind that for every bit you add, you’re doubling the scope of the puzzle. So, since we added TWO bits, we doubled, and then doubled again. This impacts not only the number of potential values, but also the number of rows, and the number of blocks.
So this is what happened:
BLOCK ONE
1 3 5 7 9 11 13 15
17 19 21 23 25 27 29 31
33 35 37 39 41 43 45 47
49 51 53 55 57 59 61 63
65 67 69 71 73 75 77 79
81 83 85 87 89 91 93 95
97 99 101 103 105 107 109 111
113 115 117 119 121 123 125 127
129 131 133 135 137 139 141 143
145 147 149 151 153 155 157 159
161 163 165 167 169 171 173 175
177 179 181 183 185 187 189 191
193 195 197 199 201 203 205 207
209 211 213 215 217 219 221 223
225 227 229 231 233 235 237 239
241 243 245 247 249 251 253 255
BLOCK TWO
2 3 6 7 10 11 14 15
18 19 22 23 26 27 30 31
34 35 38 39 42 43 46 47
50 51 54 55 58 59 62 63
66 67 70 71 74 75 78 79
82 83 86 87 90 91 94 95
98 99 102 103 106 107 110 111
114 115 118 119 122 123 126 127
130 131 134 135 138 139 142 143
146 147 150 151 154 155 158 159
162 163 166 167 170 171 174 175
178 179 182 183 186 187 190 191
194 195 198 199 202 203 206 207
210 211 214 215 218 219 223 224
227 228 231 232 235 236 239 240
243 244 247 248 251 252 254 255
BLOCK THREE
4 5 6 7 12 13 14 15
20 21 22 23 28 29 30 31
36 37 38 39 44 45 46 47
52 53 54 55 60 61 62 63
68 69 70 71 76 77 78 79
84 85 86 87 92 93 94 95
100 101 102 103 108 109 110 111
116 117 118 119 124 125 126 127
132 133 134 135 140 141 142 143
148 149 150 151 156 157 158 159
164 165 166 167 172 173 174 175
180 181 182 183 188 189 190 191
196 197 198 199 204 205 206 207
212 213 214 215 220 221 222 223
228 229 230 231 236 237 238 239
244 245 246 247 252 253 254 255
BLOCK FOUR
8 9 10 11 12 13 14 15
24 25 26 27 28 29 30 31
40 41 42 43 44 45 46 47
56 57 58 59 60 61 62 63
72 73 74 75 76 77 78 79
88 89 90 91 92 93 94 95
104 105 106 107 108 109 110 111
120 121 122 123 124 125 126 127
136 137 138 139 140 141 142 143
152 153 154 155 156 157 158 159
168 169 170 171 172 173 174 175
184 185 186 187 188 189 190 191
200 201 202 203 204 205 206 207
216 217 218 219 220 221 222 223
232 233 234 235 236 237 238 239
248 249 250 251 252 253 254 255
BLOCK FIVE
16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 31
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63
80 81 82 83 84 85 86 87
88 89 90 91 92 93 94 95
112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127
144 145 146 147 148 149 150 151
152 153 154 155 156 157 158 159
176 177 178 179 180 181 182 183
184 185 186 187 188 189 190 191
208 209 210 211 212 213 214 215
216 217 218 219 220 221 222 223
240 241 242 243 244 245 246 247
248 249 250 251 252 253 254 255
BLOCK SIX
32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47
48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63
96 97 98 99 100 101 102 103
104 105 106 107 108 109 110 111
112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127
160 161 162 163 164 165 166 167
168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183
184 185 186 187 188 189 190 191
224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239
240 241 242 243 244 245 246 247
248 249 250 251 252 253 254 255
BLOCK SEVEN
64 65 66 67 68 69 70 71
72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87
88 89 90 91 92 93 94 95
96 97 98 99 100 101 102 103
104 105 106 107 108 109 110 111
112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127
192 193 194 195 196 197 198 199
200 201 202 203 204 205 206 207
208 209 210 211 212 213 214 215
216 217 218 219 220 221 222 223
224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239
240 241 242 243 244 245 246 247
248 249 250 251 252 253 254 255
BLOCK EIGHT
128 129 130 131 132 133 134 135
136 137 138 139 140 141 142 143
144 145 146 147 148 149 150 151
152 153 154 155 156 157 158 159
160 161 162 163 164 165 166 167
168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183
184 185 186 187 188 189 190 191
192 193 194 195 196 197 198 199
200 201 202 203 204 205 206 207
208 209 210 211 212 213 214 215
216 217 218 219 220 221 222 223
224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239
240 241 242 243 244 245 246 247
248 249 250 251 252 253 254 255
Notice that the last number is 255 - this is because if all bits in an 8-bit binary number are set to 1, then the resulting decimal number is 255. This is why it’s the last number in every single block.
Hope everyone had a great Christmas and that it’s not too late to throw this puzzle in front of some relatives and blow them away with math!