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Rectangular Logic (Posted on 2005-01-28) Difficulty: 4 of 5
Promising them an increase in their allowance if they get the answer, I offer my two sons, Peter and Paul, the following puzzler:

"I am thinking of a rectangle with integer sides, each of which are greater than one inch. The total perimeter of the rectangle is no greater than eighty inches."

I then whisper the total area to Peter and the total perimeter to Paul. Neither of them are allowed to tell the other what they heard: their job is to work out the rectangle's dimensions.

Their subsequent conversation goes like this:

Peter: Hmmm... I have no idea what the perimeter is.
Paul: I knew you were going to say that. However, I don't know what the area is.
Peter: Still no clue as to the perimeter...
Paul: But now I know what the area is!
Peter: And I know what the perimeter is!

What are the dimensions of the rectangle?

No Solution Yet Submitted by Sam    
Rating: 3.7500 (12 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 18 of 44 |

At the beginning there are 361 possibilities:
        area perim 
 2 x  2;   4  8
 2 x  3;   6 10
 2 x  4;   8 12
 3 x  3;   9 12
 2 x  5;  10 14
 3 x  4;  12 14
 2 x  6;  12 16
 3 x  5;  15 16
 4 x  4;  16 16
 2 x  7;  14 18
 3 x  6;  18 18
 4 x  5;  20 18
 2 x  8;  16 20
 3 x  7;  21 20
 4 x  6;  24 20
 5 x  5;  25 20
 2 x  9;  18 22
 3 x  8;  24 22
 4 x  7;  28 22
 5 x  6;  30 22
 2 x 10;  20 24
 3 x  9;  27 24
 4 x  8;  32 24
 5 x  7;  35 24
 6 x  6;  36 24
 2 x 11;  22 26
 3 x 10;  30 26
 4 x  9;  36 26
 5 x  8;  40 26
 6 x  7;  42 26
 2 x 12;  24 28
 3 x 11;  33 28
 4 x 10;  40 28
 5 x  9;  45 28
 6 x  8;  48 28
 7 x  7;  49 28
 2 x 13;  26 30
 3 x 12;  36 30
 4 x 11;  44 30
 5 x 10;  50 30
 6 x  9;  54 30
 7 x  8;  56 30
 2 x 14;  28 32
 3 x 13;  39 32
 4 x 12;  48 32
 5 x 11;  55 32
 6 x 10;  60 32
 7 x  9;  63 32
 8 x  8;  64 32
 2 x 15;  30 34
 3 x 14;  42 34
 4 x 13;  52 34
 5 x 12;  60 34
 6 x 11;  66 34
 7 x 10;  70 34
 8 x  9;  72 34
 2 x 16;  32 36
 3 x 15;  45 36
 4 x 14;  56 36
 5 x 13;  65 36
 6 x 12;  72 36
 7 x 11;  77 36
 8 x 10;  80 36
 9 x  9;  81 36
 2 x 17;  34 38
 3 x 16;  48 38
 4 x 15;  60 38
 5 x 14;  70 38
 6 x 13;  78 38
 7 x 12;  84 38
 8 x 11;  88 38
 9 x 10;  90 38
 2 x 18;  36 40
 3 x 17;  51 40
 4 x 16;  64 40
 5 x 15;  75 40
 6 x 14;  84 40
 7 x 13;  91 40
 8 x 12;  96 40
 9 x 11;  99 40
10 x 10; 100 40
 2 x 19;  38 42
 3 x 18;  54 42
 4 x 17;  68 42
 5 x 16;  80 42
 6 x 15;  90 42
 7 x 14;  98 42
 8 x 13; 104 42
 9 x 12; 108 42
10 x 11; 110 42
 2 x 20;  40 44
 3 x 19;  57 44
 4 x 18;  72 44
 5 x 17;  85 44
 6 x 16;  96 44
 7 x 15; 105 44
 8 x 14; 112 44
 9 x 13; 117 44
10 x 12; 120 44
11 x 11; 121 44
 2 x 21;  42 46
 3 x 20;  60 46
 4 x 19;  76 46
 5 x 18;  90 46
 6 x 17; 102 46
 7 x 16; 112 46
 8 x 15; 120 46
 9 x 14; 126 46
10 x 13; 130 46
11 x 12; 132 46
 2 x 22;  44 48
 3 x 21;  63 48
 4 x 20;  80 48
 5 x 19;  95 48
 6 x 18; 108 48
 7 x 17; 119 48
 8 x 16; 128 48
 9 x 15; 135 48
10 x 14; 140 48
11 x 13; 143 48
12 x 12; 144 48
 2 x 23;  46 50
 3 x 22;  66 50
 4 x 21;  84 50
 5 x 20; 100 50
 6 x 19; 114 50
 7 x 18; 126 50
 8 x 17; 136 50
 9 x 16; 144 50
10 x 15; 150 50
11 x 14; 154 50
12 x 13; 156 50
 2 x 24;  48 52
 3 x 23;  69 52
 4 x 22;  88 52
 5 x 21; 105 52
 6 x 20; 120 52
 7 x 19; 133 52
 8 x 18; 144 52
 9 x 17; 153 52
10 x 16; 160 52
11 x 15; 165 52
12 x 14; 168 52
13 x 13; 169 52
 2 x 25;  50 54
 3 x 24;  72 54
 4 x 23;  92 54
 5 x 22; 110 54
 6 x 21; 126 54
 7 x 20; 140 54
 8 x 19; 152 54
 9 x 18; 162 54
10 x 17; 170 54
11 x 16; 176 54
12 x 15; 180 54
13 x 14; 182 54
 2 x 26;  52 56
 3 x 25;  75 56
 4 x 24;  96 56
 5 x 23; 115 56
 6 x 22; 132 56
 7 x 21; 147 56
 8 x 20; 160 56
 9 x 19; 171 56
10 x 18; 180 56
11 x 17; 187 56
12 x 16; 192 56
13 x 15; 195 56
14 x 14; 196 56
 2 x 27;  54 58
 3 x 26;  78 58
 4 x 25; 100 58
 5 x 24; 120 58
 6 x 23; 138 58
 7 x 22; 154 58
 8 x 21; 168 58
 9 x 20; 180 58
10 x 19; 190 58
11 x 18; 198 58
12 x 17; 204 58
13 x 16; 208 58
14 x 15; 210 58
 2 x 28;  56 60
 3 x 27;  81 60
 4 x 26; 104 60
 5 x 25; 125 60
 6 x 24; 144 60
 7 x 23; 161 60
 8 x 22; 176 60
 9 x 21; 189 60
10 x 20; 200 60
11 x 19; 209 60
12 x 18; 216 60
13 x 17; 221 60
14 x 16; 224 60
15 x 15; 225 60
 2 x 29;  58 62
 3 x 28;  84 62
 4 x 27; 108 62
 5 x 26; 130 62
 6 x 25; 150 62
 7 x 24; 168 62
 8 x 23; 184 62
 9 x 22; 198 62
10 x 21; 210 62
11 x 20; 220 62
12 x 19; 228 62
13 x 18; 234 62
14 x 17; 238 62
15 x 16; 240 62
 2 x 30;  60 64
 3 x 29;  87 64
 4 x 28; 112 64
 5 x 27; 135 64
 6 x 26; 156 64
 7 x 25; 175 64
 8 x 24; 192 64
 9 x 23; 207 64
10 x 22; 220 64
11 x 21; 231 64
12 x 20; 240 64
13 x 19; 247 64
14 x 18; 252 64
15 x 17; 255 64
16 x 16; 256 64
 2 x 31;  62 66
 3 x 30;  90 66
 4 x 29; 116 66
 5 x 28; 140 66
 6 x 27; 162 66
 7 x 26; 182 66
 8 x 25; 200 66
 9 x 24; 216 66
10 x 23; 230 66
11 x 22; 242 66
12 x 21; 252 66
13 x 20; 260 66
14 x 19; 266 66
15 x 18; 270 66
16 x 17; 272 66
 2 x 32;  64 68
 3 x 31;  93 68
 4 x 30; 120 68
 5 x 29; 145 68
 6 x 28; 168 68
 7 x 27; 189 68
 8 x 26; 208 68
 9 x 25; 225 68
10 x 24; 240 68
11 x 23; 253 68
12 x 22; 264 68
13 x 21; 273 68
14 x 20; 280 68
15 x 19; 285 68
16 x 18; 288 68
17 x 17; 289 68
 2 x 33;  66 70
 3 x 32;  96 70
 4 x 31; 124 70
 5 x 30; 150 70
 6 x 29; 174 70
 7 x 28; 196 70
 8 x 27; 216 70
 9 x 26; 234 70
10 x 25; 250 70
11 x 24; 264 70
12 x 23; 276 70
13 x 22; 286 70
14 x 21; 294 70
15 x 20; 300 70
16 x 19; 304 70
17 x 18; 306 70
 2 x 34;  68 72
 3 x 33;  99 72
 4 x 32; 128 72
 5 x 31; 155 72
 6 x 30; 180 72
 7 x 29; 203 72
 8 x 28; 224 72
 9 x 27; 243 72
10 x 26; 260 72
11 x 25; 275 72
12 x 24; 288 72
13 x 23; 299 72
14 x 22; 308 72
15 x 21; 315 72
16 x 20; 320 72
17 x 19; 323 72
18 x 18; 324 72
 2 x 35;  70 74
 3 x 34; 102 74
 4 x 33; 132 74
 5 x 32; 160 74
 6 x 31; 186 74
 7 x 30; 210 74
 8 x 29; 232 74
 9 x 28; 252 74
10 x 27; 270 74
11 x 26; 286 74
12 x 25; 300 74
13 x 24; 312 74
14 x 23; 322 74
15 x 22; 330 74
16 x 21; 336 74
17 x 20; 340 74
18 x 19; 342 74
 2 x 36;  72 76
 3 x 35; 105 76
 4 x 34; 136 76
 5 x 33; 165 76
 6 x 32; 192 76
 7 x 31; 217 76
 8 x 30; 240 76
 9 x 29; 261 76
10 x 28; 280 76
11 x 27; 297 76
12 x 26; 312 76
13 x 25; 325 76
14 x 24; 336 76
15 x 23; 345 76
16 x 22; 352 76
17 x 21; 357 76
18 x 20; 360 76
19 x 19; 361 76
 2 x 37;  74 78
 3 x 36; 108 78
 4 x 35; 140 78
 5 x 34; 170 78
 6 x 33; 198 78
 7 x 32; 224 78
 8 x 31; 248 78
 9 x 30; 270 78
10 x 29; 290 78
11 x 28; 308 78
12 x 27; 324 78
13 x 26; 338 78
14 x 25; 350 78
15 x 24; 360 78
16 x 23; 368 78
17 x 22; 374 78
18 x 21; 378 78
19 x 20; 380 78
 2 x 38;  76 80
 3 x 37; 111 80
 4 x 36; 144 80
 5 x 35; 175 80
 6 x 34; 204 80
 7 x 33; 231 80
 8 x 32; 256 80
 9 x 31; 279 80
10 x 30; 300 80
11 x 29; 319 80
12 x 28; 336 80
13 x 27; 351 80
14 x 26; 364 80
15 x 25; 375 80
16 x 24; 384 80
17 x 23; 391 80
18 x 22; 396 80
19 x 21; 399 80
20 x 20; 400 80


When Peter first tries, he can't find the answer, so the following unique values are eliminated by their area:

elim  2 x  2;   4  8
elim  2 x  3;   6 10
elim  2 x  4;   8 12
elim  3 x  3;   9 12
elim  2 x  5;  10 14
elim  3 x  5;  15 16
elim  2 x  7;  14 18
elim  3 x  7;  21 20
elim  5 x  5;  25 20
elim  3 x  9;  27 24
elim  5 x  7;  35 24
elim  2 x 11;  22 26
elim  3 x 11;  33 28
elim  7 x  7;  49 28
elim  2 x 13;  26 30
elim  3 x 13;  39 32
elim  5 x 11;  55 32
elim  5 x 13;  65 36
elim  7 x 11;  77 36
elim  2 x 17;  34 38
elim  3 x 17;  51 40
elim  7 x 13;  91 40
elim  2 x 19;  38 42
elim  7 x 14;  98 42
elim  3 x 19;  57 44
elim  5 x 17;  85 44
elim  9 x 13; 117 44
elim 11 x 11; 121 44
elim  5 x 19;  95 48
elim  7 x 17; 119 48
elim 11 x 13; 143 48
elim  2 x 23;  46 50
elim  6 x 19; 114 50
elim  3 x 23;  69 52
elim  7 x 19; 133 52
elim  9 x 17; 153 52
elim 13 x 13; 169 52
elim  4 x 23;  92 54
elim  8 x 19; 152 54
elim  5 x 23; 115 56
elim  7 x 21; 147 56
elim  9 x 19; 171 56
elim 11 x 17; 187 56
elim 13 x 15; 195 56
elim  6 x 23; 138 58
elim 10 x 19; 190 58
elim  5 x 25; 125 60
elim  7 x 23; 161 60
elim 11 x 19; 209 60
elim 13 x 17; 221 60
elim  2 x 29;  58 62
elim  8 x 23; 184 62
elim 12 x 19; 228 62
elim 14 x 17; 238 62
elim  3 x 29;  87 64
elim  9 x 23; 207 64
elim 13 x 19; 247 64
elim 15 x 17; 255 64
elim  2 x 31;  62 66
elim  4 x 29; 116 66
elim 10 x 23; 230 66
elim 11 x 22; 242 66
elim 14 x 19; 266 66
elim 16 x 17; 272 66
elim  3 x 31;  93 68
elim  5 x 29; 145 68
elim 11 x 23; 253 68
elim 13 x 21; 273 68
elim 15 x 19; 285 68
elim 17 x 17; 289 68
elim  4 x 31; 124 70
elim  6 x 29; 174 70
elim 10 x 25; 250 70
elim 12 x 23; 276 70
elim 14 x 21; 294 70
elim 16 x 19; 304 70
elim 17 x 18; 306 70
elim  5 x 31; 155 72
elim  7 x 29; 203 72
elim  9 x 27; 243 72
elim 11 x 25; 275 72
elim 13 x 23; 299 72
elim 15 x 21; 315 72
elim 16 x 20; 320 72
elim 17 x 19; 323 72
elim  6 x 31; 186 74
elim  8 x 29; 232 74
elim 14 x 23; 322 74
elim 15 x 22; 330 74
elim 17 x 20; 340 74
elim 18 x 19; 342 74
elim  7 x 31; 217 76
elim  9 x 29; 261 76
elim 11 x 27; 297 76
elim 13 x 25; 325 76
elim 15 x 23; 345 76
elim 16 x 22; 352 76
elim 17 x 21; 357 76
elim 19 x 19; 361 76
elim  2 x 37;  74 78
elim  8 x 31; 248 78
elim 10 x 29; 290 78
elim 13 x 26; 338 78
elim 14 x 25; 350 78
elim 16 x 23; 368 78
elim 17 x 22; 374 78
elim 18 x 21; 378 78
elim 19 x 20; 380 78
elim  3 x 37; 111 80
elim  9 x 31; 279 80
elim 11 x 29; 319 80
elim 13 x 27; 351 80
elim 14 x 26; 364 80
elim 15 x 25; 375 80
elim 16 x 24; 384 80
elim 17 x 23; 391 80
elim 18 x 22; 396 80
elim 19 x 21; 399 80
elim 20 x 20; 400 80


But since Paul states he already knew Peter would fail to find the answer on his first try, the following are eliminated as possibilities, as, for each perimeter there's at least one area that was unique. For example those with perimeter 14 are eliminated because 2x5 had been a possibility for that, and 10 was a unique area for that:

prev elim  3 x  4;  12 14
prev elim  2 x  6;  12 16
prev elim  4 x  4;  16 16
prev elim  3 x  6;  18 18
prev elim  4 x  5;  20 18
prev elim  2 x  8;  16 20
prev elim  4 x  6;  24 20
prev elim  2 x 10;  20 24
prev elim  4 x  8;  32 24
prev elim  6 x  6;  36 24
prev elim  3 x 10;  30 26
prev elim  4 x  9;  36 26
prev elim  5 x  8;  40 26
prev elim  6 x  7;  42 26
prev elim  2 x 12;  24 28
prev elim  4 x 10;  40 28
prev elim  5 x  9;  45 28
prev elim  6 x  8;  48 28
prev elim  3 x 12;  36 30
prev elim  4 x 11;  44 30
prev elim  5 x 10;  50 30
prev elim  6 x  9;  54 30
prev elim  7 x  8;  56 30
prev elim  2 x 14;  28 32
prev elim  4 x 12;  48 32
prev elim  6 x 10;  60 32
prev elim  7 x  9;  63 32
prev elim  8 x  8;  64 32
prev elim  2 x 16;  32 36
prev elim  3 x 15;  45 36
prev elim  4 x 14;  56 36
prev elim  6 x 12;  72 36
prev elim  8 x 10;  80 36
prev elim  9 x  9;  81 36
prev elim  3 x 16;  48 38
prev elim  4 x 15;  60 38
prev elim  5 x 14;  70 38
prev elim  6 x 13;  78 38
prev elim  7 x 12;  84 38
prev elim  8 x 11;  88 38
prev elim  9 x 10;  90 38
prev elim  2 x 18;  36 40
prev elim  4 x 16;  64 40
prev elim  5 x 15;  75 40
prev elim  6 x 14;  84 40
prev elim  8 x 12;  96 40
prev elim  9 x 11;  99 40
prev elim 10 x 10; 100 40
prev elim  3 x 18;  54 42
prev elim  4 x 17;  68 42
prev elim  5 x 16;  80 42
prev elim  6 x 15;  90 42
prev elim  8 x 13; 104 42
prev elim  9 x 12; 108 42
prev elim 10 x 11; 110 42
prev elim  2 x 20;  40 44
prev elim  4 x 18;  72 44
prev elim  6 x 16;  96 44
prev elim  7 x 15; 105 44
prev elim  8 x 14; 112 44
prev elim 10 x 12; 120 44
prev elim  2 x 22;  44 48
prev elim  3 x 21;  63 48
prev elim  4 x 20;  80 48
prev elim  6 x 18; 108 48
prev elim  8 x 16; 128 48
prev elim  9 x 15; 135 48
prev elim 10 x 14; 140 48
prev elim 12 x 12; 144 48
prev elim  3 x 22;  66 50
prev elim  4 x 21;  84 50
prev elim  5 x 20; 100 50
prev elim  7 x 18; 126 50
prev elim  8 x 17; 136 50
prev elim  9 x 16; 144 50
prev elim 10 x 15; 150 50
prev elim 11 x 14; 154 50
prev elim 12 x 13; 156 50
prev elim  2 x 24;  48 52
prev elim  4 x 22;  88 52
prev elim  5 x 21; 105 52
prev elim  6 x 20; 120 52
prev elim  8 x 18; 144 52
prev elim 10 x 16; 160 52
prev elim 11 x 15; 165 52
prev elim 12 x 14; 168 52
prev elim  2 x 25;  50 54
prev elim  3 x 24;  72 54
prev elim  5 x 22; 110 54
prev elim  6 x 21; 126 54
prev elim  7 x 20; 140 54
prev elim  9 x 18; 162 54
prev elim 10 x 17; 170 54
prev elim 11 x 16; 176 54
prev elim 12 x 15; 180 54
prev elim 13 x 14; 182 54
prev elim  2 x 26;  52 56
prev elim  3 x 25;  75 56
prev elim  4 x 24;  96 56
prev elim  6 x 22; 132 56
prev elim  8 x 20; 160 56
prev elim 10 x 18; 180 56
prev elim 12 x 16; 192 56
prev elim 14 x 14; 196 56
prev elim  2 x 27;  54 58
prev elim  3 x 26;  78 58
prev elim  4 x 25; 100 58
prev elim  5 x 24; 120 58
prev elim  7 x 22; 154 58
prev elim  8 x 21; 168 58
prev elim  9 x 20; 180 58
prev elim 11 x 18; 198 58
prev elim 12 x 17; 204 58
prev elim 13 x 16; 208 58
prev elim 14 x 15; 210 58
prev elim  2 x 28;  56 60
prev elim  3 x 27;  81 60
prev elim  4 x 26; 104 60
prev elim  6 x 24; 144 60
prev elim  8 x 22; 176 60
prev elim  9 x 21; 189 60
prev elim 10 x 20; 200 60
prev elim 12 x 18; 216 60
prev elim 14 x 16; 224 60
prev elim 15 x 15; 225 60
prev elim  3 x 28;  84 62
prev elim  4 x 27; 108 62
prev elim  5 x 26; 130 62
prev elim  6 x 25; 150 62
prev elim  7 x 24; 168 62
prev elim  9 x 22; 198 62
prev elim 10 x 21; 210 62
prev elim 11 x 20; 220 62
prev elim 13 x 18; 234 62
prev elim 15 x 16; 240 62
prev elim  2 x 30;  60 64
prev elim  4 x 28; 112 64
prev elim  5 x 27; 135 64
prev elim  6 x 26; 156 64
prev elim  7 x 25; 175 64
prev elim  8 x 24; 192 64
prev elim 10 x 22; 220 64
prev elim 11 x 21; 231 64
prev elim 12 x 20; 240 64
prev elim 14 x 18; 252 64
prev elim 16 x 16; 256 64
prev elim  3 x 30;  90 66
prev elim  5 x 28; 140 66
prev elim  6 x 27; 162 66
prev elim  7 x 26; 182 66
prev elim  8 x 25; 200 66
prev elim  9 x 24; 216 66
prev elim 12 x 21; 252 66
prev elim 13 x 20; 260 66
prev elim 15 x 18; 270 66
prev elim  2 x 32;  64 68
prev elim  4 x 30; 120 68
prev elim  6 x 28; 168 68
prev elim  7 x 27; 189 68
prev elim  8 x 26; 208 68
prev elim  9 x 25; 225 68
prev elim 10 x 24; 240 68
prev elim 12 x 22; 264 68
prev elim 14 x 20; 280 68
prev elim 16 x 18; 288 68
prev elim  2 x 33;  66 70
prev elim  3 x 32;  96 70
prev elim  5 x 30; 150 70
prev elim  7 x 28; 196 70
prev elim  8 x 27; 216 70
prev elim  9 x 26; 234 70
prev elim 11 x 24; 264 70
prev elim 13 x 22; 286 70
prev elim 15 x 20; 300 70
prev elim  2 x 34;  68 72
prev elim  3 x 33;  99 72
prev elim  4 x 32; 128 72
prev elim  6 x 30; 180 72
prev elim  8 x 28; 224 72
prev elim 10 x 26; 260 72
prev elim 12 x 24; 288 72
prev elim 14 x 22; 308 72
prev elim 18 x 18; 324 72
prev elim  2 x 35;  70 74
prev elim  3 x 34; 102 74
prev elim  4 x 33; 132 74
prev elim  5 x 32; 160 74
prev elim  7 x 30; 210 74
prev elim  9 x 28; 252 74
prev elim 10 x 27; 270 74
prev elim 11 x 26; 286 74
prev elim 12 x 25; 300 74
prev elim 13 x 24; 312 74
prev elim 16 x 21; 336 74
prev elim  2 x 36;  72 76
prev elim  3 x 35; 105 76
prev elim  4 x 34; 136 76
prev elim  5 x 33; 165 76
prev elim  6 x 32; 192 76
prev elim  8 x 30; 240 76
prev elim 10 x 28; 280 76
prev elim 12 x 26; 312 76
prev elim 14 x 24; 336 76
prev elim 18 x 20; 360 76
prev elim  3 x 36; 108 78
prev elim  4 x 35; 140 78
prev elim  5 x 34; 170 78
prev elim  6 x 33; 198 78
prev elim  7 x 32; 224 78
prev elim  9 x 30; 270 78
prev elim 11 x 28; 308 78
prev elim 12 x 27; 324 78
prev elim 15 x 24; 360 78
prev elim  2 x 38;  76 80
prev elim  4 x 36; 144 80
prev elim  5 x 35; 175 80
prev elim  6 x 34; 204 80
prev elim  7 x 33; 231 80
prev elim  8 x 32; 256 80
prev elim 10 x 30; 300 80
prev elim 12 x 28; 336 80


Other than those that would not have allowed Paul to determine ahead of time that Peter's first try would have been unsuccessful, the remaining choices when Paul first looks are:

remaining  2 x  9;  18 22
remaining  3 x  8;  24 22
remaining  4 x  7;  28 22
remaining  5 x  6;  30 22
remaining  2 x 15;  30 34
remaining  3 x 14;  42 34
remaining  4 x 13;  52 34
remaining  5 x 12;  60 34
remaining  6 x 11;  66 34
remaining  7 x 10;  70 34
remaining  8 x  9;  72 34
remaining  2 x 21;  42 46
remaining  3 x 20;  60 46
remaining  4 x 19;  76 46
remaining  5 x 18;  90 46
remaining  6 x 17; 102 46
remaining  7 x 16; 112 46
remaining  8 x 15; 120 46
remaining  9 x 14; 126 46
remaining 10 x 13; 130 46
remaining 11 x 12; 132 46

At this stage, as he stated, there's nothing new to his knowledge--he knew that Peter wouldn't have found anything.

Now Peter, knowing now that Paul predicted his failure on the first round, also has this reduced list, from which we can eliminate those that now have unique areas:

elim  2 x  9;  18 22
elim  3 x  8;  24 22
elim  4 x  7;  28 22
elim  4 x 13;  52 34
elim  6 x 11;  66 34
elim  7 x 10;  70 34
elim  8 x  9;  72 34
elim  4 x 19;  76 46
elim  5 x 18;  90 46
elim  6 x 17; 102 46
elim  7 x 16; 112 46
elim  8 x 15; 120 46
elim  9 x 14; 126 46
elim 10 x 13; 130 46
elim 11 x 12; 132 46

So Paul, knowing this, has left to consider

remaining  5 x  6;  30 22
remaining  2 x 15;  30 34
remaining  3 x 14;  42 34
remaining  5 x 12;  60 34
remaining  2 x 21;  42 46
remaining  3 x 20;  60 46

Of which, only 5x6 has a unique perimeter so that Paul can determine it now.


  Posted by Charlie on 2005-02-01 16:03:14
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