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).
Gauss the Mathmatician
Carl Friedrich Gauss, born in Germany in the last quarter of the C18th, was dubbed the "Prince of Mathematics.” He demonstrated his potential early, correcting his father’s arithmetic before the age of three. At the age of twelve, he questioned Euclid’s axioms. One source states that “his genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible if n is the product of distinct prime Fermat numbers” - something we all understand intuitively, right? At 24, he published Disquisitiones Arithmeticae, thought by many to be the greatest book of pure mathematics ever.
Herr Gauss published fewer papers than some other great mathematicians, and therefore others initially earned the credit for many concepts which he had actually been first to discover and confirm. Given what was learned later, he may have been the greatest theorem prover ever and, partly because of this, Gauss is “widely agreed to be the most brilliant and productive mathematician who ever lived.” He is reputed to have been the premier number theoretician of all time. His specific contributions are too numerous to list here and my guess is that most people would not even recognize the names of most of them and you can include us in that number! A thought-provoking quotation attributed to him: "It is not knowledge, but the act of learning, . . . which grants the greatest enjoyment . . .” One hopes that these puzzle caches are providing that sort of enjoyment to some who have never been exposed to the concepts illustrated therein.
Ontario’s University of Waterloo pays credit to Gauss by holding annually a mathematics contest - one of several they sponsor - named after him. It is open to all students in Grades 7 and 8 and interested (presumably very capable!) students from lower grades.
Gauss the Cache
The Essence of this Cache
There is nothing new in this puzzle cache. Its purpose is to give interested cachers the opportunity to put together what they have learned in the first six caches in this series. You will need to ensure that you recall:
- the convention for naming triangles (Archimedes);
- the Pythagorean Theorem (Bernoulli);
- working with proportions (Cayley);
- sines, cosines and tangents (Descartes);
- the use of a calculator in working with these special ratios (Fermat);
- co-ordinates from previous caches in the series.
An Example
Harold, an ardent angler, lives 320 m from the river as the crow flies but is blocked by a dense cedar bush (no bushwacking possible here without a machete or two!) from walking directly to it. He reaches the river, therefore, along a straight road which runs by his house and meets the river at an angle of 38.0°. To the nearest tenth of a metre, what distance does he have to walk to reach the river?
To do this problem:
- Make a neat diagram representing the data and you will have a right-angled triangle;
- The 320 metre distance will be the opposite side with respect to the known angle which is 38.0°;
- Place “x” on the requested (unknown) distance which will be the hypotenuse of the triangle;
- Together, the ratio of the 320 over x will be (opposite over hypotenuse) the sine of the 38° angle;
- So, we can make the statement: 320/x = sin 38 . . . the slash should be a horizontal line :-(;
- Use your knowledge of proportions to isolate x - i.e. to get a statement stating that x = something;
- In this case, that statement is: x = 320/(sin 38);
- Use your calculator to calculate this quantity and the answer is 519.76616 . . . ;
- Round off as requested and the answer is 519.8 m.
The Problems
1. Some kids in a park near a micro cache (and they hung around so long we never did get the cache!) were flying a kite at the end of a 93.0 m cord the end of which was considered to be on the ground, and it - the ground - was considered to be level! The angle between the ground and the cord was measured to be 43.0°. What was the height of the kite above the ground. (Please round your answer off to one decimal place.)
Answer: AB.C = _ _ . _ m
2. Volunteers creating a trail in the Majestic Maples Conservation Area are faced with the prospect of building a bridge across a straight section of a river and need to determine the width in order to obtain the materials necessary for the job. They marked two points - X (a post on their side) and Y (a small tree on the other side) directly opposite each other. They then measured 110 m along the river and marked point Z such that XZ was at right angles to XY. From Z, they determined that the angle between ZX and ZY was 28.9°. To the nearest tenth of a metre (i.e. one decimal place), how far is it across the river from X to Y?
Answer: DE.F = _ _ . _ m
(Friendly Advice: If problems 3 and 4 are done in steps, be sure to keep at least two decimal places in your figures - four in the case of sine, cosine and tangent - until the final answer; otherwise your rounded off answers will very likely be incorrect!)
3. From a boat 750 m out from shore, the angle of elevation of the top of the new OGA summer headquarters (comfortable but not ostentatious!) right on the shore (adjoining their spacious boathouse) near Wilberforce is 1.8° and the angle of elevation of the top of the flagpole erected on the roof is 2.5°. What is the height of the flagpole (above the level of the flat roof on which it stands) to the nearest tenth of a metre (i.e. one decimal place)? (The height above water of the observation point in the boat is irrelevant.)
Answer: G.H = _ . _ m
4. The flagpole on the flat roof of the new OGA headquarters is anchored by four cables attached at a height on the flagpole of 6.5 m and each having an angle of 30.0° with the flagpole. If 16.0 cm total - i.e. including both ends - is required for anchoring each cable, what is the total length of cable required? Round your answer off to the nearest tenth of a metre.
Answer: JK.L = _ _ . _ m
In addtion, M is the hundredths digit of the west co-ordinate of Cayley and N is the thousandths digit of the north co-ordinate of Fermat.
Finding the Co-ordinates
The co-ordinates are: N 44° ab.cde’ and W 078° fg.hjk’ where these lower case letters are unrelated to the upper case letters used to represent the answers to the problems
A = ___; B = ___; C = ___; D = ___; E = ___; F = ___; G = ___;
H = ___; J = ___; K = ___; L = ___; M = ___; N = ___.
a = L - D = ___
b = (A)(J) - G = ___
c = F - B = ___
d =C - N = ___
e =(B)(C) - (H)(J) = ___
f = D - H = ___
g =L + (D - C) = ___
h =G - A = ___
j =M + (F - A) = ___
k = B + (K)(E) = ___
You Will Find the Cache at
N 44° _ _ . _ _ _ ’ and W 078° _ _ . _ _ _ ’
Check Your Answer
You can check your answers for this puzzle on GeoChecker.com.
Other Comments
- The difficulty level has been determined by observations made by one member of the team of the other half of the team working the way through the problems!
- The cache is a “Decon” container, wired in place;
- There is little room for swag - just small articles;
- Please provide your own pen or pencil;
Cachers be warned: this place has the potential to have deep snow, to be very wet or to be insect-ridden depending on the season.