Teaser 2545: Emblematic
From The Sunday Times, 3rd July 2011 [link] [link]
Pat’s latest art installation consists of a large triangle with a red border and, within it, a triangle with a green border. To construct the green triangle, he drew three lines from the vertices of the red triangle to points one third of the way (clockwise) along their respective opposite red sides. Parts of these lines formed the sides of the green triangle. In square centimetres, the area of the red triangle is a three-digit number and the area of the green triangle is the product of those three digits.
What is the area of the red triangle?
[teaser2545]
GeoffR, and 
Frits are discussing. 
Hugo are discussing. 
Pete Good, and 
S2T2 
Jim Randell 8:46 am on 14 November 2024 Permalink |
See also: Teaser 3233, Teaser 2865, Enigma 1076, Enigma 320, Enigma 1313.
This is another puzzle that can be solved using Routh’s Theorem [@wikipedia], which I have made notes on here [ rouths-theorem.pdf ].
This Python program runs in 70ms. (Internal runtime is 3.5ms).
from enigma import (irange, inf, rdiv, multiply, nsplit, printf) # ratio XYZ/ABC in Routh's theorem def routh(x, y, z): a = x * y * z - 1 b = (x * y + y + 1) * (y * z + z + 1) * (z * x + x + 1) return rdiv(a * a, b) # compute the ratio r = green / red r = routh(2, 2, 2) # consider possible areas for the smaller (green) triangle for G in irange(1, inf): # calculate the area of the larger (red) triangle R = rdiv(G, r) # R is a 3-digit number if R > 999: break if R < 100 or R % 1 > 0: continue # the product of R's digits is G if not (multiply(nsplit(R)) == G): continue # output solution printf("G={G} R={R} [r={r}]")Solution: The area of the red triangle is: 735 cm².
And the area of the green triangle is: 105 cm².
From Routh’s Theorem we determine that area of the green triangle is 1/7 that of the red triangle.
So we are looking for a 3-digit number ABC such that:
The following run file executes in 73ms. (Internal runtime of the generated program is 144µs).
LikeLike