GAUSS’S SHORTCUT

When the great German mathematician and astronomer Carl Friedrich Gauss was a schoolboy, his teacher asked the class to add together all the whole numbers from 1 to 100. To the teacher’s astonishment, young Gauss answered immediately.

1. Can you figure out Gauss’s clever shortcut? Hint: Don’t add the numbers in their original order. Find a pattern that will make your job easier, starting from both ends and working your way back toward the middle.

2. Using the same method, add the whole numbers from 1 to 1,000.

3. Add the even numbers from 2 to 2,000.

4. Add the odd numbers from 1 to 1,001. Here’s a hint:

ADDING BACKWARD

The Guinness Book of World Records lists “math evangelist” Scott Flansburg as the fastest human calculator. His talent is special, but his methods can help everyone master the language of numbers. Try this technique, which Flansburg explains in his book, Math Magic (humancalculator.com).

In school we learn to add numbers from right to left, carrying digits to the left as necessary. This method certainly works but is prone to error, requires confusing reversals, and can lead to shaky mathematical confidence.

A better method is to add left to right. For instance, to compute the sum in the blue box below, start with the hundreds place. In the left column we have 1, 3, and 5, which stand for 100, 300, and 500. Add these up and we get 900. Already we know the final answer is at least 900, probably closer to 1,000.

In the middle column we have 2, 0, and 4, which stand for 20, 0, and 40. Add these to our running total and we get 960. So far, so good.

In the right column we have 7, 1, and 6, which total 14. Add that number to 960 and we get our answer: 974. Working left to right gets you the same answer as working right to left, but left to right makes more sense.

1. Without using paper or a calculator, add the other three sets of numbers shown above in your head, working left to right.

2. There are many other shortcuts for working with numbers. Multiplying by 10 is easy: Just tack on a zero. Want to multiply by 5? Multiply by 10 and divide by 2. Now, without using paper or a calculator, adapt these tricks to compute to the penny the standard 15 percent restaurant tip on a bill of $26.14.

SEEING DOTS

When math enthusiast Greg Tang tutored his daughter’s first grade class, he noticed that every dot in a basket of dominoes had a pencil mark on it, which meant that the kids were counting the dots one by one. But Tang knew it was quicker to add the dots in groups. Inspired by the experience, Tang started to write playful, fun-filled children’s books that teach kids problem-solving skills.

Try these counting problems based on Tang’s first book, The Grapes of Math (Scholastic Press). In each of the three exercises, add up the dots as quickly as possible without counting them one by one. What techniques make the job easier?

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