Problem Consider all integer combinations of ab for $latex 2 \leq a \leq 5 $ and $latex 2 \leq b \leq 5$: 22=4, 23=8, 24=16, 25=32 32=9, 33=27, 34=81, 35=243 42=16, 43=64, 44=256, 45=1024 52=25, 53=125, 54=625, 55=3125 If they are then placed in numerical order, with any repeats removed, we get the following sequence …
Problem A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are: 012 021 102 120 201 210 What …
Problem A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number. A number n …
Problem Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows: It can be verified that the sum of the numbers on the diagonals is 101. What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral …
Problem You are given the following information, but you may prefer to do some research for yourself. 1 Jan 1900 was a Monday. Thirty days has September, April, June and November. All the rest have thirty-one, saving February alone, which has twenty-eight, rain or shine. And on leap years, twenty-nine. A leap year occurs on …
Problem By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23. That is, 3 + 7 + 4 + 9 = 23. Find the maximum total from top to bottom in triangle.txt (right click and ‘Save Link/Target …
Problem By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23. That is, 3 + 7 + 4 + 9 = 23. Find the maximum total from top to bottom of the triangle below: NOTE: As …
Problem The Fibonacci sequence is defined by the recurrence relation: Fn = Fn-1 + Fn-2, where F1 = 1 and F2 = 1. Hence the first 12 terms will be: F1 = 1 F2 = 1 F3 = 2 F4 = 3 F5 = 5 F6 = 8 F7 = 13 F8 = 21 F9 …
Problem If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total. If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would …
Problem Starting in the top left corner of a 2×2 grid, there are 6 routes (without backtracking) to the bottom right corner. How many routes are there through a 20×20 grid? Solution It took me a while to figure out that this problem is actually a simple combination problem – consider a X by Y …
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