The Seven Musketeers | The Game of Science | EUROtoday

Our common commentator Manuel Amorós has simplified the answer to the ox downside, posed final week, in an ingenious method that may have undoubtedly happy Newton himself:

“I will propose an algebraic procedure that does not require too much thought. Let us suppose that the initial height of the grass is h and the weekly growth rate is v. We can then establish the following proportions.

10/3(h + 4v)—————12 oxen———————4 weeks

10(h + 9v)——————21 oxen———————9 weeks

24(h + 18v)——————x oxen——————–18 weeks

From the first two we obtain the relationship h = 12v

Finally we can clear x = 36

Using the same procedure, I think that in the second case 100 is the solution.”

Along the identical strains of saving psychological vitality, Susana Luu says:

“The long hand makes one revolution per hour and the short hand makes one revolution every 12 hours. Suppose 5 minutes have passed since 12 o’clock. The long hand will be at 5 minutes, but the short hand is no longer at 0 minutes, but slightly ahead, so this is not a valid result. If we allow another “almost 5 minutes” (just below 5 minutes) to cross, the lengthy hand might be nearly at 2, however the brief hand will now be behind. Again this isn’t a legitimate consequence, however assuming continuity, the brief hand, going from forward to the legitimate consequence to behind to the legitimate consequence, will in some unspecified time in the future present a legitimate consequence. It is straightforward to calculate such a time explicitly, however it isn’t essential to know that it exists. The identical reasoning applies for all subsequent 5-minute intervals, with the only exception of the 2 day by day intervals the place we arrive at 12 o’clock. Thus there might be as many legitimate moments per day as there are 5-minute intervals in 24 hours minus 2. Total: 286″.

That is, 143 each 12 hours. And Manuel Amorós clarifies:

“Each solution position is formed by the position of two needles, but the fact that the needles coincide does not imply that they are two solutions in one, it is ONE solution like any other. What does happen is that there is another solution that also results in the same position of the needles at the initial point, so in effect there are 43 solutions.” What is that different answer?

Francisco Montesinos, for his half, believes that the motion of the minute hand will not be steady (electrical clock), however strikes in leaps of 1 minute (mechanical clock), which introduces an fascinating variant (see feedback from final week).

Since we have now solved (or not less than tried to) a number of algebraic issues recently, it’s a good time to keep in mind that, when transferring from elementary arithmetic to algebra, the standard “four operations” are normally now not enough. Just as an addition with the added finish repeated is a multiplication, a multiplication with the issue repeated is an influence. And simply as addition has its reverse operation in subtraction and multiplication in division, exponentiation has its reverse in root extraction, that’s, acquiring sq. roots, cubic roots, and many others. That's six operations already… What's the seventh? (Even when you don't keep in mind it, you possibly can deduce it by extrapolating from the above).

While you consider it, listed below are three little issues concerning the sixth operation:

Which is bigger, the fifth root of 5 or the sq. root of two?

Which is bigger, the fourth root of 4 or the seventh root of seven?

Which is bigger, √7 + √10 or √3 + √19?

In all three circumstances, it’s a matter of discovering the answer mentally (or, when you can’t do with out paper and pencil, by performing solely quite simple arithmetic operations).