Problem Some positive integers n have the property that the sum [ n + reverse(n) ] consists entirely of odd (decimal) digits. For instance, 36 + 63 = 99 and 409 + 904 = 1313. We will call such numbersreversible; so 36, 63, 409, and 904 are reversible. Leading zeroes are not allowed in either …
Problem The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145: 1! + 4! + 5! = 1 + 24 + 120 = 145 Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to …
Problem Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4. Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6. Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal. What …
Problem By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles: Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution. Solution This problem looks more difficult than it is, I’m not going to go …
Problem In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to 2427. Find the minimal path sum, in matrix.txt (right click and ‘Save Link/Target As…’), a 31K text file …
Problem Comparing two numbers written in index form like 211 and 37 is not difficult, as any calculator would confirm that 211 = 2048 < 37 = 2187. However, confirming that 632382518061 > 519432525806 would be much more difficult, as both numbers contain over three million digits. Using base_exp.txt (right click and ‘Save Link/Target As…’), …
Problem In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way: High Card: Highest value card. One Pair: Two cards of the same value. Two Pairs: Two different pairs. Three of a Kind: Three cards of the same value. Straight: All cards are …
Problem The rules for writing Roman numerals allow for many ways of writing each number (see FAQ:Roman Numerals). However, there is always a “best” way of writing a particular number. For example, the following represent all of the legitimate ways of writing the number sixteen: IIIIIIIIIIIIIIII VIIIIIIIIIII VVIIIIII XIIIIII VVVI XVI The last example being …
Problem Consider quadratic Diophantine equations of the form: x2 – Dy2 = 1 For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 1. It can be assumed that there are no solutions in positive integers when D is square. By finding minimal solutions in x for D = {2, 3, …
Problem Let p(n) represent the number of different ways in which n coins can be separated into piles. For example, five coins can separated into piles in exactly seven different ways, so p(5)=7. OOOOO OOOO O OOO OO OOO O O OO OO O OO O O O O O O O O Find the …
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