The time has come to issue a warning: this entire series of caches will be archived between May 1st and October 31st, 2022.
Caches in the Series
The series consists of: Archimedes (GC3TT8C), Bernoulli (GC3V9MW), Cayley (GC3W898), Descartes (GC3X0W7), Euclid (GC3XGN0), Fermat (GC413V5), Gauss (GC4219Z), Heron (GC4VGBX), Ingham (GC4WC72), Jarnik (GC4Z9PK), Kepler (GC4ZMRG), Lovelace (GC50BN6), Merryman (GC51D34) and Newton (GC5AJ04).
Equipment
For this cache, you will find it helpful to use a straight edge for drawing - a 15-cm ruler is perfect - and you should have access to some sort of scientific calculator whether it be on the computer, on your “smart ‘phone” or one of those ten-dollar portable items we used at school.
Euclid the Mathematician
Euclid of Alexandria, who bridged the third and fourth centuries B.C., is known as the most eminent mathematician of the remote past - even if we know very little about him. He is most famous for his having produced The Elements, a publication of thirteen “books” on plane (flat surface) geometry, number theory, irrational numbers and three-dimensional geometry. Sixty years ago, high school students in Ontario spent an entire year of their schooling in mathematics based on the early books - learning the “propositions” and using them to solve “deductions.” Euclid’s work was impressive in its progressive nature. He began with postulates such as “it is possible to draw a straight line between any two points” and axioms such as “things which are equal to the same thing are equal to each other.” Having accepted these “unnecessary-to-prove” statements as his foundations, he meticulously amassed, brick by brick, an impressive body of proofs. Working one’s way through parts of this work brings delight to some and night-mares to others but none could fail to be impressed by the sequential nature of proofs each building on those which came before.
Indicative of the respect in which Euclid is held is the fact that the University of Waterloo in Waterloo, Ontario, Canada, holds an annual mathematics contest bearing his name. Eligible students are those in their “final year of secondary school . . . and motivated students in lower grades.”
Euclid, the Geocache
A Few Words of Introduction
The first caches in this series have been based on puzzles involving the proper naming of triangles, the Pythagorean Theorem, properties of similar triangles, proportions and the definitions of the special ratios - sine, cosine and tangent. In this cache we provide nothing new - simply some “real life” problems based on the information from the previous four caches. As always, we urge cachers to make neat diagrams - they often help! And don't forget that calculator! Good luck!
The “Puzzles”
1. (Pythagorean Theorem) A delightful trail through mature mixed forest around Lake Teriskawa is interrupted by a marsh which occupies a bay at the north-west “corner" of the lake. Here, one is forced to take to the road and walk 300 m north before turning east and walking a further 199 m to the point where one can enter the continuation of the trail. Some members of the “Friends of the Teriskawa Trail” - wish to build a boardwalk to bridge this gap. If a straight boardwalk were constructed to fill the space - from trail-end to trail-head, so to speak - what would be the resulting reduction (to the nearest metre) in walking distance on the trail compared to what it is at the moment? (Your three-digit answer will provide the A, B and C below.)
2. (Similar Triangles) Burj Khalifa is a rather tall building in Dubai! Paul and Elaine had been informed of its height but decided to test it - and their precision in measuring! - for themselves. They recorded the co-ordinates at the base of the structure and then, at a time of day when the top of the shadow of the building fell on an accessible place on the ground, recorded the co-ordinates there as well. From the co-ordinates, they were able to determine the length of the building’s shadow as 471 metres. At the same time, they measured the length of the shadow of a metre stick and found it to be 57 cm. Knowing that the sun is so far distant that its rays make the same angle with any objects reasonably close together on the globe’s surface, they were able to calculate the height of the Burj Khalifa. What height did they find it to be correct to the nearest metre? (Your three-digit answer will provide the D, E and F below.)
3. (Similar Triangles) An observer whose height is considered to be zero (don’t worry about ego problems here - the observer is fictional!) observes the top of a mountain, known to be a horizontal distance of 2.4 km away, to be in line with the tip of a 22.8 m high church spire 30 m away - also measured horizontally. How high is the mountain to the nearest metre? (Your four-digit answer will provide G, H, T and J below.)
4. (Proportions) On Friday, Fred and Sue set out from Moretonhampstead to cycle to the Bellever youth hostel on Dartmoor (in Devon, England) which they reached not quite as early as anticipated because of the caching they did along the way. Having spent a weekend there hiking and caching in the area with other hostellers, they completed their trip across Dartmoor reaching the Exeter hostel late on Monday afternoon. Surprised at the length of time the journey had taken - even with the caching along the way - they examined their map - an Ordnance Survey publication with a scale of 1:50 000 - and determined that they had travelled a distance of 73.6 cm on the map. To the nearest kilometre, how far had they cycled across the moor from Moretonhampstead to Exeter? (Your two-digit answer will provide the K and L below.)
5. (Not a “real life” problem - more a “can you interpret the description?” type of problem!) ABCD is a four-sided closed figure (i.e. a “quadrilateral”) and convention dictates that the letters go around the figure - either clockwise or counter-clockwise - in the order given. B is joined to D forming a right angle, ∢ABD, and ∢BCD is also a right angle. Sides BC, CD and AD are 4, 3 and 13 cm respectively (i.e. in the order given). In fractional form, what are:
(a) sin ∢CDB? (M/N below)
(b) cos ∢BDA? (O/P below)
(c) tan ∢BAD? (Q/R below)
The Co-ordinates of the Cache
The co-ordinates are: N 44° ab.cde’ and W 078° fg.hij where the lower case letters are unrelated to any letters previously mentioned on this cache page. It is important to note that S is the hundredths digit of the minutes part of the north co-ordinate of the location of the cache in “Descartes.”
A = ___; B = ___; C = ___; D = ___; E = ___;
F = ___; G = ___; H = ___; T = ___; J = ___;
K = ___; L = ___; M = ___; N = ___; O = ___;
P = ___; Q = ___; R = ___; S = ___.
a = L - H + T: ____
b = P - D + B: ____
c = R - S - E: ____
d = N + M - F: ____
e = J + K + A:____
f = C + G - Q: ____
g = (D)(E) - (B + C): ____
h = 3H – 4F: ____
i = (3P + A)/(3J - L): ____
j = (7G - 3T) - (3M - 5K): ____.
The co-ordinates of the cache are: : N 44° __ __ . __ __ __ ’ and W 078°__ __ . __ __ __ ‘.
Other Notes
- Please provide your own pen or pencil.
- The cache container is a clean camouflaged peanut butter jar.
- The cache is located along a rural trail where the space and sky are major reasons for being there.
- There is some minor bushwacking involved at the end of the search.
Check Your Answer
You can check your answers for this puzzle on GeoChecker.com.