Wikipedia:Reference desk/Archives/Mathematics/2014 August 26

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August 26

Commutative magmas?

Have commutative magmas been studied? Is there something like partially associative (and everywhere commutative) magma like there is a trace monoid (for partially commutative but everywhere associative magma)? JMP EAX (talk) 11:53, 26 August 2014 (UTC)

Derive variable values from formulae

Given the following 9 formulae with known solutions, is it possible to derive the values of the other variables?

• A = B + C + D
• E = 2(F + G) + C
• H = I + J + K
• L = B + I + J
• M = C + D + J
• N = O + C + D + K
• P = C + I + J
• Q = B + C + I + J
• R = S + B + D + J

To be clear, the values are known for A, E, H, L, M, N, P, Q, and R. Is it possible to discover the values of any other variable for certain? ΣΑΠΦ (Sapph)^Talk 13:03, 26 August 2014 (UTC)

Sure. Note that F and G only occur together, so you can replace F + G with some new variable T. Now you've got 9 equations and 9 unknowns, so in general you'd expect to be able to solve for the unknowns using Gaussian Elimination or a similar technique. And indeed, you can in this case. So the only values you can't get are F and G, but you can at least get F + G.--80.109.106.3 (talk) 13:19, 26 August 2014 (UTC)

If you only want to know whether you can discover at least one value then consider Q-P. PrimeHunter (talk) 13:46, 26 August 2014 (UTC)

As a point of clarification, it is not necessarily the case that any assignment of values to the variables listed will lead to solutions for the whole system. The system may be inconsistent. In the vein of PrimeHunter's suggestion, you can also see by inspection that Q-L=C. SemanticMantis (talk) 19:46, 26 August 2014 (UTC)

This particular set of equations however is consistent and has a solution for any values of A, E, H, L, M, N, P, Q, and R (and you can choose any value for either F or G too). --catslash (talk) 21:28, 26 August 2014 (UTC)

You have 1 free variable (G). The solutions for your other variables are: B = Q - P, C = Q - L, D = A + L + P - 2*Q, F = E/2 + L/2 - G - Q/2, I = A + L - M - 2*Q + 2*P, J = M + Q - A - P, K = H + Q - L- P, O = L + N - A - H, S = P + R - L - M 70.190.182.236 (talk) 22:28, 26 August 2014 (UTC)

If you want a little fun, solve MINUSONE ZERO FIFTH QUARTER THIRD HALF ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN ELEVEN TWELVE THIRTEEN FOURTEEN FIFTEEN SIXTEEN SEVENTEEN NINETEEN TWENTY TWENTYONE TWENTYTWO TWENTYTHREE TWENTYFOUR TWENTYFIVE TWENTYSIX TWENTYSEVEN TWENTYEIGHT TWENTYNINE Where each "word" adds the relevant unknowns to get the right number (ie T+E+N = 10)

-- 82.44.187.221 (talk) 08:48, 27 August 2014 (UTC)

Let's see how far we can get by inspection on this...

extended content
The following discussion has been closed. Please do not modify it.
SIXTEEN= SIX + TEN, so TEEN=TEN, so E=0 THIRTEEN=THREE + TEN = THR + TN, so I=0 NINE = 2N = 9, so N = 4.5 TEN = T+4.5 = 10, so T=5.5 FIVE + TEN = FIFTEEN, so FV=FF=5, so F=V=2.5 ELEVEN = L+2.5+4.5=11, so L=11-7=4 ONE = O+4.5=1, so O=-3.5 TWO = 5.5+W-3.5, so W=0 TWENTY = 5.5+4.5+5.5+Y = 15.5+Y, so Y=4.5 SEVEN = S+2.5+4.5 = S+7, so S=0 SIX=X=6 FIFTH = 2.5+2.5+5.5+H = 10.5+H, so H=-10.3 THREE = 5.5-10.3+R = -4.8+R =3, so R=7.8 FOUR = 2.5-3.5+U+7.8 = 8.8+U=4, so U=4.8 (sanity check!)TWELVE = 5.5+0+0+4+2.5=12 (so it seems to actually be consistent!) EIGHT = G-10.3+5.5 = G-4.8=8, G=3.2 HALF = -10.3+A+4+2.5 = -3.8+A, so A=4.3 And the rest are trivial, since M, Q and Z only appear in one number each

Turns out it is consistent, and we can get all the way (hatted to keep it short, and in case anyone else wants a try) MChesterMC (talk) 08:37, 28 August 2014 (UTC)

And next week, we'll prove that N=1 implies P=NP. - ¡Ouch! (^{hurt me} / _{more pain}) 09:56, 1 September 2014 (UTC)