The “spring check” in 2019 will be the last of the regular formal checks; thereafter, we will visit a cache only when we are informed in a log or personal e-mail that maintenance is needed - so please keep us informed!
Equation Solving
Compass is the third in a series of fifteen puzzle caches of increasing difficulty based on solving equations. It is recommended that one start with Abacus (GC6A006) and continue in alphabetical sequence through Brackets (GC6AR81), Compass (GC6AWD0), Decagon (GC6C4EE), Ellipse (GC6CFW3), FOIL (GC6HDNE), Googol (GC751JK), Hectare (GC75BD3), Index (GC75EAM), Jargon (GC75X8W), Kite (GC76010), LessThan (GC763HV ), Median (GC7DH17), Nonagon (GC7DWC4) and Obelus (GC7DWCD).
The Name - Compass - Nothing to do with the Puzzle!
Definition of a Compass (once called “compasses”)
A compass is a device with two arms - usually about ten centimetres in length - attached at one end such that if desired, the arms can be made to move apart in one plane creating different sized angles between them. One arm ends in a sharp metal point and the other with a device made to hold a pencil so that the device can be used to draw arcs or circles.
We just spent an hour with a compass, two sharp pencils, a straight edge and lots and lots of paper doing things we hadn’t thought of for years. Among the numerous things one can do with this simple equipment are:
- use its name for a cache when you need a mathematical term starting with "c;"
- draw a perfect ninety degree angle to a given line from any given point;
- find the point which would be the centre of a circle drawn precisely through three given points;
- draw an angle exactly the same size as another angle;
- draw a line which divides an angle precisely in half;
- find the exact middle of a line segment;
- draw a circle inside a triangle which will just touch each of the sides at one point precisely;
- create interesting (especially if they are then coloured) patterns with arcs and circles . . . and on and on.
Simplifying Equations to Solve Them by Inspection
Prerequisites
It is assumed that you have some skill at solving equations or have done Abacus and Brackets the GC identifications for which are at the top of the page. We have chosen equations in which we ask you to simplify each side. Once that is done correctly, each should be easily solved by inspection.
Examples as Explanations
(1) 4x - 5 + x + 5 - 4x = 2x - 9 - 2x + 11
”Collecting “like terms” on each side yields:
4x + x - 4x - 5 + 5 = 2x - 2x - 9 + 11
Completing the simplification yields:
x = 2 . . . so the equation is “solved!”
(2) 2x - 5 - 6a + 3 + 5x + 6a + 2 = 3x2 + 8 + 3 - 3x2 + 6 - 4 + 7 + 1
Collecting the like terms on each side of the equals sign yields:
7x + 0 + 0a = 21 + 0x2
Normally, we would not bother writing the terms which have 0 as the numerical coefficient because each equals zero, so we now have:
7x = 21 and thus, by inspection,
x = 3
(3) r2 + 9 - 3 r2 + 6 + 2r2 = - 4 + 2d + 7 - 4d - 3 + 3d
Collecting the like terms on each side yields:
15 = d or, as is the accepted form,
d = 15
(4) For this example, remember that / indicates division so, for 2x/3, make a fraction with 2x in the numerator (top) and 3 in the denominator (bottom).
2x/3 - 3/4 + 1x/3 + 3/4 = 1p/2 - 2 - 8 - 1p/2 + 3 + 4
which simplifies to:
3x/3 + 0/4 = 0p/2 - 3
which would normally be written:
x = - 3 . . . and you’re done!
Now . . . the Puzzle Questions!
Simplify each side of each equation and you should find that you have an equation for which the solution is easily spotted by “inspection.” As you work, please note that in the solving of equations, no equals sign is used at the beginning of a line as it is when simplifying expressions.
(1)
4x - 5 - 3x + 5 = 2x - 1 - 2x + 3
(2)
5a + 7 - 3a - 7 = - 4a - 2 + 4a + 12
(3)
7 + 5b/3 - 4b/3 - 7 = 2b/5 - 2 - 2b/5 + 5
(4)
- 5y + 5 + 6y - 8 = 4y - 3 + 11 - 4y
(5)
5c - 2 + (3c - 1 - 7c + 5) = (5c - 3 - 4c + 5) - c + 7
(6)
10 + 4d3 - 7 - 4d3 = 12 - 2d2 + 5d + 2d2 - 4d - 12
(7)
5e + 1 - 2e + 4 - 3e = 8 - 6e + 7e - 4
(8)
(- 5f + 7 - 3f) + (13 + 8f + 4) = 4 + 10f + (1 - 6f - 5)
(9)
2g2 + 7 - 5g2 - 2 + 3g2 - 4 = - 3g3 + 2g - 4 - g + 3g3 - 3
(10)
3h/4 + 2h2/3 - 1 - 2h2/3 + 3 - 3h/4 = 7 + 4h/5 - 7 - 3h/5
The Co-ordinates of the Cache
The co-ordinates are:
N 44° L M . N P Q’ and W 078° R S . T U W’
where the capital letters needed can be calculated by substituting below using the solutions you have just obtained and, for your convenience, those answers can be filled in here.
a = ___ ; b = ___ ; c = ___ ; d = ___ ; e = ___ ;
f = ___ ; g = ___ ; h = ___ ; x = ___ ; y = ___ .
* * * * * * * * * * * * * * * * * * * * * * * * * * *
L = y - h = ___ ; M = x + f = ___ ; N = c - d = ___ ; P = b - e = ___ ;
Q = y - a = ___ ; R = h - f = ___ ; S = g - (c + e) = ___ ;
T = b - (x + c) = ___ ; U = a + g - (d + x) = ___ ; W = h - (b - g) = ___ .
So you are looking for:
N 44° __ __ . __ __ __ ‘ and W 078° __ __ . __ __ __ ‘.
Comments
- Please provide your own writing utensil;
- There will be some water in some seasons - but there is an indirect path available;
- Tweezers might be needed.
- You can check your answers for this puzzle on GeoChecker.com.