Lead Pencils : a big waste!

One repeats the following process with lead pencils.

A. Sharpen to a cone pointed tip.

B. Flatten the head through use.

So if you divide the "cylindrical" lead structure in the pencil along the length into mini cylinders. Then you basically remove 67% the volume by sharpening the mini cylinder and using the conic remainder.

So you lose at least 67% of the lead in the pencil by sharpening. You can save unnecessary waste of wood and lead by not using lead pencils.

TIP: A cylinder has 3 times the volume of a cone of the same base and height.

can never divide?

1. Given that x is a positive integer prove that f(x) = x² + x + 1 will never divide by 5.

2. Consider the expression x^x + 1, where x be a positive integer.

It can be verified that x = 7 is the least value for which x^x + 1 divides by 2³.

Given that n is a positive integer, find the least value of x for which x^x + 1 is divisible by 2ⁿ.

convolutions and sum of sums

Let S_k be the sequence of simple sums, that is S₁=1, S₂ =1+2, S_k =1+2+...+k then

S₁ + S₂ + ... + S_k = k.1+(k-1).2+(k-2).3+ ... + 2.(k-1)+1.k = the convolution of the sequence {1, ...,k}.

In your spare time you can show that sum of sums equals (n)(n+1)(n+2)/(1.2.3) and sum of sum of sums equals n(n+1)(n+2)(n+3)/(1.2.3.4) and so on.

1.2.3 = 6.
1.2.3.4 = 24.

Fibonacci Sums

The Fibonacci series is: F₁=1, F₂=2, for n=2,3,... F_n+1= F_n + F_n-1. The (n+1)-st number is the sum of n-th and (n-1)-th number in the sequence.

Fibonacci numbers have the property that: the sum of the first n numbers of a sequence is contained in the sequence. Do you know of others?

F₁+F₂+F₃+...+F_n = F_n+2-1

Actually the sequence G₁=1, G₂=2,for n=2,3,... G_n+1= G_n + ... + G₂ + G₁ trivially staifies that property. So, Fibonacci sequence is not unique in the above sense. Can you think of a sequence {H_k} such that

H₁+H₂+H₃+...+H_n = H_n+3 - c? for some constant c.

On a tangent: Kolmogorov information complexity speaks of representations that can compress information effectively. Not only does the following set of characters "F₁=1, F₂=2 F_n+1= F_n + F_n-1" contain the entire Fibonacci sequence, but also the sum of its first n elements.