Sunday Times Teaser 3178 – Drying Boards

by BRG on August 18, 2023

by Mark Valentine

Published Sunday August 20 2023 (link)

Chef Ignacio liked to prop his two identical thin rectangular chopping boards against the shelf at the end of his counter to dry. He placed the top of the first one flush with the shelf corner and rested the second on the first, as shown in the diagram. To aid drying, he positioned the second to maximise the air volume in the bounded region below it. The length of each board is an even number of cm (less than 26cm) [1]

The distance between the bases of the two boards was a whole number of mm. What is this distance?

[1] and the height of the shelf above the counter is an integer number of millimetres.

From → Uncategorized

3 Comments Leave one →

BRG permalink


# see: https://sundaytimesteasers.files.wordpress.com/2023/08/st3178-9.pdf
# for details. To maximise the area under the second board the triangle it
# forms must be isosceles:
#    __
#       |\
#       |t\
#       |  \       cos(t) = (c/2) / d )
#       |   \                         )  ==>  d = c.sqrt(c /(2.(c + b)))
#    d  |    \     cos(2.t) = b / c   )
#       |     \ c  
#       |      \           
#    __ |       \ 
#       |\__ d   \         
#       |2t \__   \        
#       |       \__\  
#    b  |       c   \__    
#       |              \__
#       |                 \__
#    __ |____________________\
#                  a

# primitive Pythagorean triples with square hypotenuses < 260
pt = {5:(7, 24, 25), 13:(119, 120, 169)}

for cr, (a, b, c) in pt.items():
  for a, b in ((a, b), (b, a)):
    # 2.(c + b) must be a square
    dn = 2 * (c + b)
    if (dr := int(dn ** (1 / 2))) ** 2 == dn:
      # scale the primitive triangle up by the denominator
      a, b, c, d = a * dr, b * dr, c * dr, c * cr
      if c % 20 == 0 and c < 260:
        print(f"{d = }mm ({a = }mm, {b = }mm, {c = }mm).")

# see: https://sundaytimesteasers.files.wordpress.com/2023/08/st3178-9.pdf

# for details. To maximise the area under the second board the triangle it

# forms must be isosceles:

# __

# |\

# |t\

# | \ cos(t) = (c/2) / d )

# | \ ) ==> d = c.sqrt(c /(2.(c + b)))

# d | \ cos(2.t) = b / c )

# | \ c

# | \

# __ | \

# |\__ d \

# |2t \__ \

# | \__\

# b | c \__

# | \__

# __ |____________________\

# a

# primitive Pythagorean triples with square hypotenuses < 260

pt = {5:(7, 24, 25), 13:(119, 120, 169)}

for cr, (a, b, c) in pt.items():

for a, b in ((a, b), (b, a)):

# 2.(c + b) must be a square

dn = 2 * (c + b)

if (dr := int(dn ** (1 / 2))) ** 2 == dn:

# scale the primitive triangle up by the denominator

a, b, c, d = a * dr, b * dr, c * dr, c * cr

if c % 20 == 0 and c < 260:

print(f"{d = }mm ({a = }mm, {b = }mm, {c = }mm).")

Frits permalink

Nice diagram.

Reply
Frits permalink

We also have:

d^2 / c^2 = h / a where h is the altitude of the isosceles triangle.

So the height of the shelf above the counter must be a multiple of a square number, a >= 4.

Reply

Sunday Times Teaser 3178 – Drying Boards

by Mark Valentine

Published Sunday August 20 2023 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta