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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)

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Solution My solution | Comment 8 of 44 |
I get the answer: area = 16, perimeter = 16, as follows: 
 
After Peter's first statement, the only possible areas he could have are those with multiple perimeters:

area: 12 (perimeters:  16 14)
area: 16 (perimeters:  20 16)
area: 18 (perimeters:  22 18)
area: 20 (perimeters:  24 18)
area: 24 (perimeters:  28 22 20)
area: 28 (perimeters:  32 22)
area: 30 (perimeters:  34 26 22)
area: 32 (perimeters:  36 24)
area: 36 (perimeters:  40 30 26 24)
area: 40 (perimeters:  44 28 26)
area: 42 (perimeters:  46 34 26)
area: 44 (perimeters:  48 30)
area: 48 (perimeters:  52 38 32 28)
area: 50 (perimeters:  54 30)
area: 52 (perimeters:  56 34)
area: 54 (perimeters:  58 42 30)
area: 56 (perimeters:  60 36 30)
area: 60 (perimeters:  64 46 38 34 32)
area: 64 (perimeters:  68 40 32)
area: 66 (perimeters:  70 50 34)
area: 68 (perimeters:  72 42)
area: 70 (perimeters:  74 38 34)
area: 72 (perimeters:  76 54 44 36 34)
area: 76 (perimeters:  80 46)
area: 45 (perimeters:  36 28)
area: 63 (perimeters:  48 32)
area: 75 (perimeters:  56 40)
area: 78 (perimeters:  58 38)
area: 81 (perimeters:  60 36)
area: 84 (perimeters:  62 50 40 38)
area: 90 (perimeters:  66 46 42 38)
area: 96 (perimeters:  70 56 44 40)
area: 99 (perimeters:  72 40)
area: 102 (perimeters:  74 46)
area: 105 (perimeters:  76 52 44)
area: 108 (perimeters:  78 62 48 42)
area: 80 (perimeters:  48 42 36)
area: 88 (perimeters:  52 38)
area: 100 (perimeters:  58 50 40)
area: 104 (perimeters:  60 42)
area: 112 (perimeters:  64 46 44)
area: 120 (perimeters:  68 58 52 46 44)
area: 128 (perimeters:  72 48)
area: 132 (perimeters:  74 56 46)
area: 136 (perimeters:  76 50)
area: 140 (perimeters:  78 66 54 48)
area: 144 (perimeters:  80 60 52 50 48)
area: 110 (perimeters:  54 42)
area: 130 (perimeters:  62 46)
area: 135 (perimeters:  64 48)
area: 150 (perimeters:  70 62 50)
area: 160 (perimeters:  74 56 52)
area: 165 (perimeters:  76 52)
area: 170 (perimeters:  78 54)
area: 175 (perimeters:  80 64)
area: 126 (perimeters:  54 50 46)
area: 156 (perimeters:  64 50)
area: 162 (perimeters:  66 54)
area: 168 (perimeters:  68 62 58 52)
area: 180 (perimeters:  72 58 56 54)
area: 192 (perimeters:  76 64 56)
area: 198 (perimeters:  78 62 58)
area: 204 (perimeters:  80 58)
area: 154 (perimeters:  58 50)
area: 182 (perimeters:  66 54)
area: 189 (perimeters:  68 60)
area: 196 (perimeters:  70 56)
area: 210 (perimeters:  74 62 58)
area: 224 (perimeters:  78 72 60)
area: 231 (perimeters:  80 64)
area: 176 (perimeters:  60 54)
area: 200 (perimeters:  66 60)
area: 208 (perimeters:  68 58)
area: 216 (perimeters:  70 66 60)
area: 240 (perimeters:  76 68 64 62)
area: 256 (perimeters:  80 64)
area: 225 (perimeters:  68 60)
area: 234 (perimeters:  70 62)
area: 252 (perimeters:  74 66 64)
area: 270 (perimeters:  78 74 66)
area: 220 (perimeters:  64 62)
area: 260 (perimeters:  72 66)
area: 280 (perimeters:  76 68)
area: 300 (perimeters:  80 74 70)
area: 264 (perimeters:  70 68)
area: 286 (perimeters:  74 70)
area: 308 (perimeters:  78 72)
area: 288 (perimeters:  72 68)
area: 312 (perimeters:  76 74)
area: 324 (perimeters:  78 72)
area: 336 (perimeters:  80 76 74)
area: 360 (perimeters:  78 76)

 
After Paul's first statement, the only perimeters possible for him are those with multiple areas, each of which correspond to one of the areas just mentioned:

perimeter: 16 (areas:  12 16)
perimeter: 18 (areas:  18 20)
perimeter: 20 (areas:  16 24)
perimeter: 22 (areas:  18 24 28 30)
perimeter: 24 (areas:  20 32 36)
perimeter: 26 (areas:  30 36 40 42)
perimeter: 28 (areas:  24 40 45 48)
perimeter: 30 (areas:  36 44 50 54 56)
perimeter: 32 (areas:  28 48 60 63 64)
perimeter: 34 (areas:  30 42 52 60 66 70 72)
perimeter: 36 (areas:  32 45 56 72 80 81)
perimeter: 38 (areas:  48 60 70 78 84 88 90)
perimeter: 40 (areas:  36 64 75 84 96 99 100)
perimeter: 42 (areas:  54 68 80 90 104 108 110)
perimeter: 44 (areas:  40 72 96 105 112 120)
perimeter: 46 (areas:  42 60 76 90 102 112 120 126 130 132)
perimeter: 48 (areas:  44 63 80 108 128 135 140 144)
perimeter: 50 (areas:  66 84 100 126 136 144 150 154 156)
perimeter: 52 (areas:  48 88 105 120 144 160 165 168)
perimeter: 54 (areas:  50 72 110 126 140 162 170 176 180 182)
perimeter: 56 (areas:  52 75 96 132 160 180 192 196)
perimeter: 58 (areas:  54 78 100 120 154 168 180 198 204 208 210)
perimeter: 60 (areas:  56 81 104 144 176 189 200 216 224 225)
perimeter: 62 (areas:  84 108 130 150 168 198 210 220 234 240)
perimeter: 64 (areas:  60 112 135 156 175 192 220 231 240 252 256)
perimeter: 66 (areas:  90 140 162 182 200 216 252 260 270)
perimeter: 68 (areas:  64 120 168 189 208 225 240 264 280 288)
perimeter: 70 (areas:  66 96 150 196 216 234 264 286 300)
perimeter: 72 (areas:  68 99 128 180 224 260 288 308 324)
perimeter: 74 (areas:  70 102 132 160 210 252 270 286 300 312 336)
perimeter: 76 (areas:  72 105 136 165 192 240 280 312 336 360)
perimeter: 78 (areas:  108 140 170 198 224 270 308 324 360)
perimeter: 80 (areas:  76 144 175 204 231 256 300 336)
 
After Peter's next statement, the only areas possible for him are those of multiple perimeters, each of which correspond to one of the perimeters just mentioned:

area: 16 (perimeters:  20 16)
area: 18 (perimeters:  22 18)
area: 20 (perimeters:  24 18)
area: 24 (perimeters:  28 22 20)
area: 28 (perimeters:  32 22)
area: 30 (perimeters:  34 26 22)
area: 32 (perimeters:  36 24)
area: 36 (perimeters:  40 30 26 24)
area: 40 (perimeters:  44 28 26)
area: 42 (perimeters:  46 34 26)
area: 44 (perimeters:  48 30)
area: 48 (perimeters:  52 38 32 28)
area: 50 (perimeters:  54 30)
area: 52 (perimeters:  56 34)
area: 54 (perimeters:  58 42 30)
area: 56 (perimeters:  60 36 30)
area: 60 (perimeters:  64 46 38 34 32)
area: 64 (perimeters:  68 40 32)
area: 66 (perimeters:  70 50 34)
area: 68 (perimeters:  72 42)
area: 70 (perimeters:  74 38 34)
area: 72 (perimeters:  76 54 44 36 34)
area: 76 (perimeters:  80 46)
area: 45 (perimeters:  36 28)
area: 63 (perimeters:  48 32)
area: 75 (perimeters:  56 40)
area: 78 (perimeters:  58 38)
area: 81 (perimeters:  60 36)
area: 84 (perimeters:  62 50 40 38)
area: 90 (perimeters:  66 46 42 38)
area: 96 (perimeters:  70 56 44 40)
area: 99 (perimeters:  72 40)
area: 102 (perimeters:  74 46)
area: 105 (perimeters:  76 52 44)
area: 108 (perimeters:  78 62 48 42)
area: 80 (perimeters:  48 42 36)
area: 88 (perimeters:  52 38)
area: 100 (perimeters:  58 50 40)
area: 104 (perimeters:  60 42)
area: 112 (perimeters:  64 46 44)
area: 120 (perimeters:  68 58 52 46 44)
area: 128 (perimeters:  72 48)
area: 132 (perimeters:  74 56 46)
area: 136 (perimeters:  76 50)
area: 140 (perimeters:  78 66 54 48)
area: 144 (perimeters:  80 60 52 50 48)
area: 110 (perimeters:  54 42)
area: 130 (perimeters:  62 46)
area: 135 (perimeters:  64 48)
area: 150 (perimeters:  70 62 50)
area: 160 (perimeters:  74 56 52)
area: 165 (perimeters:  76 52)
area: 170 (perimeters:  78 54)
area: 175 (perimeters:  80 64)
area: 126 (perimeters:  54 50 46)
area: 156 (perimeters:  64 50)
area: 162 (perimeters:  66 54)
area: 168 (perimeters:  68 62 58 52)
area: 180 (perimeters:  72 58 56 54)
area: 192 (perimeters:  76 64 56)
area: 198 (perimeters:  78 62 58)
area: 204 (perimeters:  80 58)
area: 154 (perimeters:  58 50)
area: 182 (perimeters:  66 54)
area: 189 (perimeters:  68 60)
area: 196 (perimeters:  70 56)
area: 210 (perimeters:  74 62 58)
area: 224 (perimeters:  78 72 60)
area: 231 (perimeters:  80 64)
area: 176 (perimeters:  60 54)
area: 200 (perimeters:  66 60)
area: 208 (perimeters:  68 58)
area: 216 (perimeters:  70 66 60)
area: 240 (perimeters:  76 68 64 62)
area: 256 (perimeters:  80 64)
area: 225 (perimeters:  68 60)
area: 234 (perimeters:  70 62)
area: 252 (perimeters:  74 66 64)
area: 270 (perimeters:  78 74 66)
area: 220 (perimeters:  64 62)
area: 260 (perimeters:  72 66)
area: 280 (perimeters:  76 68)
area: 300 (perimeters:  80 74 70)
area: 264 (perimeters:  70 68)
area: 286 (perimeters:  74 70)
area: 308 (perimeters:  78 72)
area: 288 (perimeters:  72 68)
area: 312 (perimeters:  76 74)
area: 324 (perimeters:  78 72)
area: 336 (perimeters:  80 76 74)
area: 360 (perimeters:  78 76)
 
After Paul's final statement, the only perimeter possible for him is the one of only one area, one of the areas just mentioned:

perimeter: 16 (area:  16)

 
After Peter's final statement, the only area possible for him is the one of that perimeter:

area: 16 (perimeter:  16)
 
 
 

 

Edited on January 31, 2005, 1:29 pm
  Posted by Penny on 2005-01-31 13:28:00

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