## Scale Balance

Saw some comments show up recently on our first scale balance puzzle (*hard!*), which reminded me how long its been since we put one of these up. With that in mind:

How many triangles are need to balance scale 3?

*feel free to submit your answers in the comment section below. Will unmask Thursday, thanks.*

falwan| Profile April 1st, 2008 - 10:28 pmOne !

TopSmart| Profile April 1st, 2008 - 11:22 pmone triangle

runninfool| Profile April 1st, 2008 - 11:33 pmComparing the first two pictures, 1 triangle = 2 circles = 4 squares since 1 circle = 2 squares.

So, the last picture has 2 squares and 1 circle which can be translated to 4 squares.

We need 1 triangle to balance this.

bobg| Profile April 2nd, 2008 - 12:14 amOne Triange

______________

X – Triange

Y – Square

Z – Circle

1. X+2Y = 3Z

2. X+Z = 6Y

rearranged to find Triagle

3. X=3Z-2Y

4. X=6Y-Z

Solve for Z

5. 3Z-2y = 6Y-Z

6. 4Z=8Y

7. Z=2Y

Check 7. with scale 1.

1. X+2y = 3Z

2. X = 2Z

SO: X = 2Z= 4Y

Therefore:

Z+2y = ?

Z + Z = ?

2Z = X

QED

Ari| Profile April 2nd, 2008 - 2:18 amThe weights are: box 1, ball 2, triangle 4

So one triangle should do the trick. Four boxes or 2 balls would work also.

Norway| Profile April 2nd, 2008 - 3:24 amTriangle= 4

Square = 1

Circle = 2

So on the first balance scale its 6-6

second is 6-6

and on the third its 4-4

So you will need a triangle on the third..

gunturi| Profile April 2nd, 2008 - 4:11 amIt is one triangle

From second balance one ball is equal to two squares.

So, one Triangle is equal to four squares.

For the third balance,

the total weight in the left hand side is 4 squares which is equal to one triangle.

vikram.mankal| Profile April 2nd, 2008 - 5:28 amLet x = Green triangle

y = Blue square

z = Orange circle

x + 2y = 3z -> (1)

x + z = 6y -> (2)

Eqn. (1) + (2) gives

2x + 2y + z = 3z + 6y

Therefore 2x = 2z + 4y

Or x = z + 2y ( Dividing throughout by 2 )

z + 2y is 1 Orange circle and 2 Blue squares

So to balance this we need

1 x or

1 Green triangle

T| Profile April 2nd, 2008 - 7:41 amAfter reading how the previous ones were done I’ve come up with…

In Balance 1, 1triangle (t) and 2 squares (2s) = 3circles (3c)

In Balance 2, 1triange (1) and 1 circle(c) = 6squares (6s)

and in Balance 3, 1 circle and 2 squares = x triangles (x(t))

Re-arranging B1 and B2 to make t the subject…

t = 3c – 2s and t = 6s – c

.: 3c – 2s = 6s – c

.: 4c = 8s

.: c = 2s

B3 can then be expressed as 2s + 2s = x(t)

.: x(t) = 2c

If 1 circle is equal to 2 squares you can take 1 circle from the right of B1 and 2 squares from the left of B1 and it will remain even.

.: 1t = 2c (Or 1 triangle balances with 2 circles)

Match this up with B3 and you get x = 1

.: it will take 1 triangle to balance scale 3

Testing with numbers… if 1t = 2c and 1c = 2s, then t and c must be even numbers. s=1, c=2 and t=4…

Balance 1 is 4+1+1 = 2+2+2 (correct)

Balance 2 is 4+2 = 1+1+1+1+1+1 (correct)

Balance 3 is 2+1+1 = 4 (correct)

Which confirms that 1 triangle will balance scale 3

joe| Profile April 2nd, 2008 - 8:21 am1 Triangle will balance the scale.

Using the information given for Triangles (T), Squares (S) and Circles (C).

First scale T + 2S = 3C

Second one T + C = 6S which rearranging tells us that C=2S (one circle is 2 squares), and using that info and reinputing in the above that T=4S.

On the left of the 3rd scale there is 1 C + 2 S or also equiv 2S + 2S =4 squares which we calculated was 1 triangle

da Moose| Profile April 2nd, 2008 - 10:06 am1 triangle = 2 circles = 4 squares

Therefore, on balance no. 3, 1 circle + 2 squares = 1 triangle

SolarFlare| Profile April 2nd, 2008 - 10:42 amOnly one green triangle is required on the rhs of balance 3. Explanation is as follows.

Scale equations are (using the first letters of Green, Blue & Amber):

G + 2B = 3A (B1)

G + A = 6B (B2)

2B + A = nG (B3)

Rearranging:

G = 3A – 2B (B1)

G = 6B – A (B2)

Subtracting these (B1 – B2) & rearranging:

4A = 8B

A = 2B

We can now change the eqns B1 & B2 to eliminate A & express G in terms of B:

3A – 2B = 6B – 2B = 4B = G (B1)

6B – A = 6B – 2B = 4B = G (B2)

(identical, as expected)

Lhs of balance 3 becomes:

2B + A = 2B + 2B = 4B

And we know from above that:

4B = G

So only one G is required on the rhs of B3.

foger1979| Profile April 2nd, 2008 - 11:25 am1 triangle

GreekMind| Profile April 2nd, 2008 - 11:30 amtriangle = t

square = s

circle = c

1t + 2s = 3c

1t + 1c = 6s

1c + 2s = ??*t

(1) => 1t + 2s = 3c => t = 3c – 2s (a)

(2) => 1t + 1c = 6s (a),(2)=> 3c – 2s +1c = 6s => 4c = 8s

so c = 2s (a) => t = 3c – 2s = 3*2s -2s = 4s

(3) 1c + 2s = 2s + 2s = 4s

and 1t = 4s

so the solution is 1 triangle

Kira| Profile April 2nd, 2008 - 11:51 amI got one triangle to balance the circle and two squares

t=triangle, s=square and c=circle

so from scale 1 we get the formula t+2s=3c and from scale 2 we have t+c=6s. From there we make t the subject of both formulas and get (Scale 1) t=3c-2s and (Scale 2) t=6s-c

joining these two equations i get

3c-2s=6s-c

3c+c=6s+2s

4c=8s

c= 2s and conversely s=c/2

if you then resubstitute these into the orginal scale 1 and scale 2 formulas you get

Scale 1

t+2s=3c

t+2s=3(2s)

t+2s=6s

t=4s or s=t/4

Scale 2

t+c=6s

t+c=6(c/2)

t+c=3c

t=2c or c=t/2

Scale 3 is c+2s, which is (in terms of t) = t/2+2(t/4)= t/2+t/2 = t

I’m sure i could’ve simplified that, couldn’t I?

ekphron| Profile April 2nd, 2008 - 11:59 am2 squares equal 1 circle

2 circles equal 1 triangle

the answer —

1 triangle is needed to balance scale 3

aaronlau| Profile April 2nd, 2008 - 12:57 pmAdding scale 1 & 2:

2 Triangles + 2 squares + 1 circle = 6 squares + 3 circles

=> 2 Triangles = 4 squares + 2 circles

=> 1 Triangle = 2 square + 1 circle ! Scale 3

suineg| PUZZLE MASTER | Profile April 2nd, 2008 - 1:38 pmI this this can be an answer:

From the first 2 scales: 1 circle= 2 squares;

1 triangle= 4 squares–> 1 triangle= 2 circles

So in the thid scale: 1 circle + 2 squares= 4 squares—–>

1 triangle, that I think is the answer

Birdieguy| Profile April 2nd, 2008 - 2:45 pmOne triangle.. unless I made a mistake in my algebra

jibbo| Profile April 2nd, 2008 - 3:35 pm1

one ball equels 2 squares

one triangle equels 4 squares

therefore 2 squares and one ball(2 squares) equels one triangle

muteddragon| Profile April 2nd, 2008 - 3:38 pmone triangle.

Poynter-Jones| Profile April 2nd, 2008 - 5:36 pmI just did a quick trial and error with the sequences of shapes and came up with this code:

2 Blue squares = 1 Orange circle

2 Orange circles = 1 green triangle

1 Green triangle = 4 Blue squares

3.)

2 blue squares = 1 orange circle;

Which leaves two orange circles, which equal one green triangle.

Therefore to balance the other side of the scale, place one green triangle.

nitwit88| Profile April 2nd, 2008 - 6:44 pm1 triangle.

circles->c, triangles->t, and squares->s

3c=1t+2s and c+t=6s. Combining these and removing t, we get that 8s=4c, or 2s=c so we can consider the left side of scale three to be two circles. Plugging that into t+2s=3c, we get t+c=3c, which yields t=2c. That’s exactly how much we have on the left, so one triangle will balance it out.

LadyInsomnia| Profile April 2nd, 2008 - 8:53 pmOne triangle is needed to balance scale 3.

Start by adding two triangles to the left-hand side. Then we have a triangle-plus-circle set and a triangle-plus-two-squares set. From scale one, we can add three circles to the right-hand side; from scale two, we can add six squares to the right-hand side, so scale three is balanced. Next, “cancel out” one circle and two squares from both sides. We’re left with two triangles balancing two sets of circle-plus-two-squares. So one triangle balances one circle plus two squares.

jasc| Profile April 2nd, 2008 - 9:25 pm1

http://img510.imageshack.us/im.....no3nw1.jpg

paprika60| Profile April 2nd, 2008 - 9:45 pmOkay, I’ve come up with ..

1Triangle=2 Circles

1Square=.5Circles

1Circle=2Squares

1Triangle=4Squares

So, 1 Triangle would balance?

tad| Profile April 3rd, 2008 - 12:26 pmOne Triangle.

1T + 2S = 3C

1T + 1C = 6S

1C + 2S = 1T

NyteAzul| Profile April 3rd, 2008 - 1:12 pmOnly 1 Triangle.

1T + 2S = 3C

-1T + 6S = 1C

8S = 4C => 2S = 1C

1T + 1C = 6S => 1T = 4S

2S + 1C = 1T

hex| PUZZLE MASTER | Profile April 3rd, 2008 - 1:33 pmScale 1: 2 x S + T = 3 x C

Scale 2: T + C = 6 x S

Solving for C and S wrt T:

C=T/2

S=T/4

Scale 3: 2 x S + C = T/2 + T/2 = T

So one triangle balances scale 3

Shawn| PUZZLE GRANDMASTER | Profile April 3rd, 2008 - 3:55 pm1 triangle.

1 triangle = 4 squares = 2 circles

RK| Profile April 4th, 2008 - 4:44 pm1 Triangle = Answer. LOTS of good explanations above for this one, check them out

michaelc| Profile April 8th, 2008 - 4:39 pmjasc,

That’s a neat way to look at it!

I got the same answer as you did, but had to do all the algebra with pencil and paper.

I’ll have to look at simpler ways to do things in the future. A picture really says a 1,000 words!

wdrudder| Profile February 12th, 2011 - 10:53 pmOne triangle. One triangle equals two circles, one circle equals two squares, one triangle equals four squares.

neutrondisciple| Profile July 5th, 2011 - 9:32 pm1st scale – 1triangle=2circles & 2squares=1circle

2nd scale – 1triangle=4squares & 1circle=2squares

3rd scale – 1circle=half a triangle & 2squares=half a triangle

so my answer is, one!

skyadz| Profile August 31st, 2011 - 10:59 pmthere is only one triangle needed to balance scale 3.

1 circle = 2 square

1 triangle = 2 circle = 4 square

to confirm..

T=triangle S=square C=circle

@ scale1

1T + 2S = 3 C

subs. T=2C S=1C

2C + 1C = 3C

3C=3C

balanced…..

@ scale2

1T + 1C = 6S

subs. T=4S C=2S

4S + 2S = 6S

6S = 6S

balanced…,

@ scale3

1C + 2S=?

T=2C T=4S,therefore C=1/2T S=1/4T

1(1/2)T + 2(1/4)T

T=1

to balanced scale 3 we need 1 triangle…,

soumen023| Profile April 24th, 2013 - 3:52 am1