A 9x9 grid is filled with numbers from 1 to 81, each number occurring exactly once. The grid is said to be ultramagic if the product of numbers in row k is equal to the product of numbers in column k for each k. Can you construct an ultramagic grid?
- via AMS puzzle corner

## Monthly Archives for February, 2014

## Talmud problem

Joe owes $100 to creditor A, $200 to creditor B and $300 to creditor C. Joe dies, leaving behind an estate. Unfortunately, Joe's estate has insufficient funds to repay his creditors (yes, that's a very small estate). The question is how does one split the estate among the contestants. Here are three situations and the action by a certain judge in each situation:
1. If the …

**Continue reading the story**"Talmud problem"## Removing balls

You have a jar with 20 black balls and 16 white balls. At each step,
1. You remove two balls.
2. If the two balls are of the same color, you add a black ball. Otherwise you add a white ball.
You repeat these steps until there is only 1 ball remaining in the jar. What is the color of the remaining ball?
As a follow-up question, suppose you …

**Continue reading the story**"Removing balls"## Coin Problem

Suppose you have an infinite stock of $a bills and $b bills such that g.c.d(a,b)=1. Find the largest amount of money (integer) that cannot be represented using $a and $b denominations.
- via Wolfram Mathematica

## Checkers

There are 3n checkers: n red, n green, n blue. They are shuffled and placed on an n x 3 board. Your objective is to rearrange the checkers so that each of the n columns has one each of red, green and blue checkers. However, you are allowed to swap checkers only if they are in the same row. Is it possible to achieve your …

**Continue reading the story**"Checkers"## An auction

A gift card of $5 is auctioned between two people. The rules are strange. The players are allowed to bid forever until both of them are unwilling to bid higher. However, each player has to pay the highest bid that (s)he made whether or not he wins the $5 gift card. Of course the gift card goes to the higher bidder. Will the bidding ever …

**Continue reading the story**"An auction"## Number placement

You are given a string of the form "?<?>?>?<...<?", containing n '?' symbols. Give an algorithm to replace each '?' by a unique integer from 1 to n such that all the inequalities in the string are satisfied. For example, if the given string is "?<?>?>?", your algorithm may return "2<4>3>1", but may not return "4<3>2>1".
- via Algorithmic puzzles

## What is the n-th number?

Suppose a sequence of natural numbers satisfies the following:
1. The sequence is strictly increasing.
2.
3. If m,n are relatively prime, .
Show that the only such sequence is .
- via Mind your decisions

## A Geometry puzzle

Find the area of a single triangle in the figure below (all triangles are right-angled).
Hint: If you use geometry rather than algebra, you will find this a challenging puzzle. That's why there are 4 triangles. Try to rearrange them.
- via Mind your decisions