Sometimes a good diagram can make all the difference. Consider this problem, ```Duke the dog's head is 6 inches long, his tail is as long as his head plus half the length of his body, and his body is as long as the head and the tail combined. ``` which can make your head hurt just reading it. But with the right diagram, ``` H = 6 +----+ T = H + B/2 | | ------------- +---------------+ +----- | | | | +---------------+ B = H + T ``` you're halfway to the solution. Here's another example where a diagram changes something confusing ```What day of the week was yesterday if 5 days before the day after tomorrow was Wednesday? ``` into something straightforward: ``` +----------+----------+---------+-----+--------+---------+ |Day before|Day before|Yesterday|Today|Tomorrow|Day after| |day before|yesterday | | | |tomorrow | |yesterday | | | | | | +----------+----------+---------+-----+--------+---------+ ^ | |________ 5 days before the day after tomorrow ``` Or consider this problem, ```Every day at noon a ship leaves San Francisco for Tokyo, and at the same instant a ship leaves Tokyo for San Francisco. Each trip lasts exactly eight days. How many Tokyo ships will each San Francisco ship meet? ``` and this diagram: ``` Day |/ / / / / / / / | 0 X / / / / / / / * |\ / / / / / / / /| | X / / / / / / / | |/ \ / / / / / / / | 1 * X / / / / / / * | / \ / / / / / / /| | / X / / / / / / | |/ / \ / / / / / / | 2 * / X / / / / / * | / / \ / / / / / /| | / / X / / / / / | |/ / / \ / / / / / | 3 * / / X / / / / * | / / / \ / / / / /| | / / / X / / / / | |/ / / / \ / / / / | 4 * / / / X / / / * | / / / / \ / / / /| | / / / / X / / / | |/ / / / / \ / / / | 5 * / / / / X / / * | / / / / / \ / / /| | / / / / / X / / | |/ / / / / / \ / / | 6 * / / / / / X / * | / / / / / / \ / /| | / / / / / / X / | |/ / / / / / / \ / | 7 * / / / / / / X * | / / / / / / / \ /| | / / / / / / / X | |/ / / / / / / / \| 8 + / / / / / / / X | / / / / / / / /| SF Tokyo ``` The right diagram can let you replace calculation with counting, or some other simple observation.