Copyright ©1997 by Paul Niquette. All rights reserved. 
he
sophisticated solver of this
puzzle would surely
start out by letting x represent the
unknown number.
At this point we do not know much about x  not even how many digits x has. For now, let's just assign n to represent the number of digits in x. Like all the rest of the cyphers in x, the least significant digit is unknown. Better give it a name: d.
With so little known, a problem like this can give you a brain blister. efore clicking off to some other website, though, let's write an equation.
Step 1. Pick a value for digit d,That Step 4 is a bit tricky. During the long division, we have to keep sticking in nines as we go along until the quotient comes out even (zero remainder, as they used to teach fourth graders before calculators). There is just this one more thing we know: That we have to get n digits in the quotient.
All we really don't know about the numerator at this point is the length of that numerator  how many 9s there are between the 8 over on the left and that 1 there on the right.
Historical Note The exhilaration of that moment gave at least a transient significance to an otherwise undistinguished life. It was truly a time to cry out, "Hoohah."
"It's my father's first name," I answered. "He inspired me to study numbers." Epilog:
For a sequel to this puzzle see Reaman Numerals
(plural). Also, in 2007, the author found out
that another name has
come along over the intervening 54 years for integers
with this feature:
"parasite
numbers."
ntuition plays a speaking role in solving problems. Thus, the word 'seems' belongs in the script and gets uttered by a whole troupe of repertory actors on the mathematical stage. After a solution is found, though, analytic purists get embarrassed about any seeming that might have gone on when the problem was not yet solved (Eeoo, the sword). When the curtain first parted on this problem back in 1953, an empty theatre echoed with an opening soliloquy from an offstage narrator, "It seems like the least significant digit in x would have to be 9." The sentiments in that line derived, no doubt, from a venerable number play of about the same vintage (see Secret Message). "Hark!" cried the Voice of Reason from the front line of the chorus. "The leftmost digit of x cannot be a zero." The chorus made sounds of agreement. "Otherwise the expression 'ndigit number' makes no sense: Why, if zero counted as a leading digit, any number of them might then be stuck on so that even the smallest integer would say of itself, 'I have an infinite number of digits'!" The audience laughed at the absurdity. The 'great unraveler,' George Polya (How to Solve It), strolled onto the stage, "So what if you cannot immediately conquer the general problem!" exclaimed he. "Be not ashamed to attack a few specific cases." Out stepped a real ham, Mr. Empirical,
grinning. He commenced
one of his 'trialanderror' schticks...
"There, you see," commented Polya. "The ratio 9 is indeed feasible. Mathematicians call that an 'existence theorem' and  " "Watch this," interrupted Mr.
Empirical...
"Seems like d has to be
9, doesn't it,"
mused the narrator, emphasizing the sword. {Return}
In slow motion, then...




