A largely shaped threat!
In a 2D universe far, far, away from here, only three types of beings thrived the lands. They were Squares (shaped like squares), Triangles (shaped like triangles), and Circles (...circles).
In a peaceful day, an anonymous group of beings went to a bank and blackmailed the cashier (believe me, the cashier was very sharp ), threatening to kill him and steal the money from the bank if he didn't answer their question.
The question goes as follows:
"In our group, more than three of us are circles, less than six of us are triangles, and seven of us are squares. We in total have more than fifteen of us, and less than eighteen of us. We in total have fourty-five sides . Tell us how many of us are there, separating each type."
Would care to help?
The answer format goes like this:
Total Shapes - Triangles - Squares - Circles
For example, if there are two triangles, five squares and one circles, the answer is typed as:
8251
S i d e N o t e
Circles have one side, squares have four sides, and triangles have three sides
The answer is 16475.
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4 solutions
Oh!
I can't express my thankfulness!
Thanks a lot for spending time on this question!
Same solution, but better presentation! Nice!
No. of shapes = 1 6 or 1 7
7 are squares.
⟹ 9 or 1 0 other shapes.
⟹ Sides used = 2 8
Left to be used = 4 5 − 2 8 = 1 7
No. of circles > 3
No. of triangles < 7
Now, by substitution,
No. of circles = C
No. of triangles = T
C + T = 9 or 1 0
C + 3 T = 1 7
For integer values of C and T ,
C = 5 and T = 4 .
⟹ C + T = 9
⟹ C + T + S = 1 6
⟹ Answer = 1 6 4 7 5
Simple, yet efficient solution. Thanks for posting it @Vinayak Srivastava !
Let C , T , S denote the number of Circles, Triangles and Squares respectively. Then according to question :
C ≥ 4
0 ≤ T ≤ 5
S = 7
1 6 ≤ C + T + S ≤ 1 7
⇒ 9 ≤ C + T ≤ 1 0 Eq. 1
4 S + 3 T + C = 4 5
⇒ 3 T + C = 4 5 − 4 ⋅ 7 = 1 7 Eq. 2
Taking C = 4 . Then according to Eq. 1, T = 5 . But these will not satisfy Eq. 2.
Taking C = 5 . Then T is either 4 or 5 . Taking T = 4 we see that Eq. 2 is also satisfied. We also see that no other pair of ( C , T ) is possible to satisfy all the constraints.
Hence, S = 7 , T = 4 , C = 5 with total number of shapes = 1 6
So the answer is 1 6 4 7 5
Thanks a lot for trying my question!
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I thought circles don't have sides.
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Sorry @Vikram Karki , I can't help on that. I gave an example too, if you weren't aware of it...
Plus, circles always have one curved side.
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@A Former Brilliant Member – But thanks to you, I updated the question by mentioning how many sides each shape has.
Let the number of circles be 3 + x , number of triangles be 6 − y where x , y > 0 . Then
1 5 < 3 + x + 6 − y + 7 < 1 8 ⟹ y − 1 < x < y + 2 .
Also, 3 + x + 3 ( 6 − y ) + 4 × 7 = 4 5 ⟹ 3 y − x = 4 ⟹ y − 1 < 3 y − 4 < y + 2 ⟹ 3 < 2 y < 6 .
Since y is an integer, therefore y = 2 ⟹ x = 3 × 2 − 2 = 2 . So, the number of circles is 3 + 2 = 5 , number of triangles is 6 − 2 = 4 , and the total number is 5 + 4 + 7 = 1 6 .
Hence the required answer is 1 6 4 7 5 .
😭😭😭 i got the correct answer but wrote how many shapes instead of how many sides
T 1 2 3 4 5 C 1 4 1 1 8 5 2 Total 7 + 1 5 = 2 2 7 + 1 3 = 2 0 7 + 1 1 = 1 8 7 + 9 = 1 6 7 + 7 = 1 4
1 6 − 4 − 7 − 5 ⟹ 1 6 4 7 5