the 'Lenny Conundrum' authors wrote:Suppose you had a life-size map of Neopia, made of paper. The thickness of the paper is 0.1 millimetres.
Now, suppose you fold it in half. Then you fold it in half again, and again, and again. In all, you fold it in half 40 times. How thick would the folded paper be?
Please give your answer in metres, round to the nearest metre, and exclude any punctuation or extra information.
(Is the above "Round 155", or have I lost count again?)
The thickness of one sheet is given as being 0.1 millimeters (mm). Upon folding once, this is doubled, to 0.2 mm. Upon folding twice, this is doubled again, to 0.4mm. The sequence of thicknesses is geometric, with a = 0.1 and common ratio (constant multiplier) r = 2. Then:
. . . . .no foldings: a[0] = 0.1
. . . . .1 folding: a[1] = 0.1(2^1) = 0.2
. . . . .2 foldings: a[2] = 0.1(2^2) = 0.4
And so forth, forming the sequence 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, ....
. . . . .n foldings: a[n] = 0.1(2^n)
(Note: The "boxed" numerals after the a's would, had we the formatting capability, be subscripts.)
Then clearly we are looking for the value of a[40], the thickness after forty folds. Following the pattern, we get:
. . . . .40 foldings: a[40] = 0.1(2^40)
Recalling that there are 1000 mm in one metre, we obtain a value of about 109951162.8 metres in thickness. Rounded to the nearest metre, we get a value of:
. . . . .109951163
This confirms the results posted earlier.
Eliz.
