By James Matteson

1 Diophantine challenge, it's required to discover 4 affirmative integer numbers, such that the sum of each of them will probably be a dice. resolution. If we suppose the first^Cx3^)/3-), the second^^x3-y3--z* ), the third=4(-z3+y3+*'), and the fourth=ws-iOM"^-*)5 then> the 1st additional to the second=B8, the 1st extra to the third=)/3, the second one further to third=23, and the 1st extra to the fourth=ir therefore 4 of the six required stipulations are chuffed within the notation. It is still, then, to make the second one plus the fourth= v3-y3Jrz*=cnbe, say=ic3, and the 3rd plus the fourth^*3- 23=cube, say=?«3. Transposing, we need to get to the bottom of the equalities v3--£=w3--if=u?--oi?; and with values of x, y, z, in such ratio, that every can be more than the 3rd. allow us to first get to the bottom of, generally phrases, the equality «'-}-23=w3-|-y3. Taking v=a--b, z=a-b, w-c--d, y=c-d, the equation, after-dividing by means of 2, turns into a(a2-)-3i2)==e(c2-J-3f72). Now think a-Sn])--Smq, b=mp-3nq, c=3nr

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1 Diophantine challenge, it's required to discover 4 affirmative integer numbers, such that the sum of each of them will likely be a dice. answer. If we think the first^Cx3^)/3-), the second^^x3-y3--z* ), the third=4(-z3+y3+*'), and the fourth=ws-iOM"^-*)5 then> the 1st further to the second=B8, the 1st further to the third=)/3, the second one extra to third=23, and the 1st extra to the fourth=ir therefore 4 of the six required stipulations are happy within the notation.

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**Example text**

20 Mathematics-Its Magic & Mastery Numbers in the three-system are added Thus: ' 2,122 212 121 1 + 2 + 2 = 12. 1 + 2 + 1 + 2 = 20. 2 + 1 + 2 + 1 = 20. 2 + 2 = 11. Write Write Write Write In the same way. 2 and carry 1. 0 and carry 2. 0 and carry 2. 11. and the sum is 11,002. The check shows that 2,122 corresponds to 71 in the decimal system, 212 corresponds to 23, and 121 corresponds to 16. 71 + 23 + 16 = 110, and 110 is 11,002 in the three-system. Below are examples of addition in various systems: Four-System Five-System Six-System 3231 133 312 11002 4312 432 243 11042 45312 5423 355 55534 Seven-System Eight-System Nine-System 56543 3635 216 64030 64753 2567 471 70233 784521 63677 2467 861776 Twelve-System 5e4t2eO 4ett2t ettee 6548019 The reader may check the sums by the decimal system.

A number trick similar to those above may be added to our repertory. Multiplication of a five-place number by 99,999 is done in the same way as a four-place number by 9,999. The resulting number trick follows the usual pattern. A Puzzler with 111,111 It will be recalled that r1mltiplication of three-place numbers by 1,001 results in the writing of this number twice, and immediately note that 111,111 must have 1,001 as a factor. Thus 111,111 = 111' 1,001, but 111 = 3' 37, and 1,001 = 7' 11' 13. Therefore 111,111 = 3' 7 .

The word million is just a little more than four hundred years old. It first appeared in Italy about 1500, eight years after Columbus discovered America. s!