SPOJ: 8060. Cookies Piles

The kids in my son’s kindergarten made Christmas cookies with their teacher, and piled them up in columns. They then arranged the columns so that the tops of the columns, going from shortest to tallest, were in a nice straight ramp. The cookies were all of uniform size. Given that there were A cookies in the shortest pile, that the difference in height between any two adjacent piles was D cookies, and that there were N piles, can you write a program to figure out how many cookies there were in total?

Problem: Sphere Online Judge (SPOJ) – Problem AMR10F

Solution: This is quite a nice and simple problem. The easiest solution is just to write a loop to calculate the amount of cookies. If you think one step further you can write a general formula for calculating this.
Let’s imagine three piles of cookies.

At the rightmost there are A cookies. The middle one got A + D cookies and the leftmost got A + 2D cookies. If we generalize this we get: \text{allCookies} = (A + 2 * D) + (A + 1 * D) + (A + 0 * D) which can be written as \sum_{i=0}^{N-1} (A + iD).
And thus can be simplified:

$latex \sum_{i=0}^{N-1} A + D \sum_{i=0}^{N-1} i = NA + D\frac{(N-1)N}{2}.

SPOJ: 4301. Tables

Byteman works as a carpenter. He has just received an order for s pine-wood tables. Although he has plenty of pine-wood boards in his workshop, he has just run out of screws. Therefore he needs to walk to the warehouse and bring back some boxes with screws. What is the minimum number of boxes that he needs to bring in order to have enough screws to make the tables?

Problem: Sphere Online Judge (SPOJ) – Problem AE1B

Solution: That’s a nice application for an greedy algorithm, i.e. you grab the largest values successively. This yields the optimal solution in this case because we don’t need to find exactly X screws. It’s sufficient to find at least X screws.

SPOJ: 1030. Triple Fat Ladies

Pattern Matchers have been designed for various sorts of patterns. Mr. HKP likes to observe patterns in numbers. After completing his extensive research on the squares of numbers, he has moved on to cubes. Now he wants to know all numbers whose cube ends in 888.

Given a number k, help Mr. HKP find the kth number (indexed from 1) whose cube ends in 888.

Problem: Sphere Online Judge (SPOJ) – Problem EIGHTS

Solution: Yeah, an other number problem! And again I just looked for patterns.

Which gets us:

EDIT: Thanks to the anonymous person who found a simple connection between the numbers but can’t properly communicate. You can find the k-th number with this simple formula: 192 + 250(k-1)

You should see that each number ends in 2. If you look closer you see that the last two digits are 42 if i is even, and 92 if i is odd. Let’s look at the other digits. We get:

If you do some basic arithmetics you see that:

You can calculate your first digits either recursively or search for a function. This takes probably some time, so I looked this sequence up. OEIS gives us the formular: a(n) = floor((5n-2)/2) for n > 2

 

SPOJ: 3442. The last digit

Nestor was doing the work of his math class about three days but he is tired of make operations a lot and he should deliver his task tomorrow. His math’s teacher gives two numbers a and b. The problem consist in find the last digit of the potency of base a and index b. Help Nestor with his problem. You are given two integer numbers: the base a (0 <= a <= 20) and the index b (0 <= b <= 2,147,483,000). You have to find the last digit of a^b.

Problem: Sphere Online Judge (SPOJ) – Problem LASTDIG

Solution: I don’t really have an background in number theory, so I decided to approach this problem empirically. My main source of information was this small Python program:

I checked the last digit for each base (0..20) and looked for patterns. Happily, I found them. Here’s a snippet:

The rest is straight forward and thanks to the 700B source code size limitations, the source code is pretty unreadable. ;)