Financially well-informed readers may have recently learned about a money-saving trick that uses a formula supposedly devised by Carl Friedrich Gauss, the 19th century maths colossus. The viral 100 envelope challenge is based on an apocryphal story that Gauss, when a young boy, outwitted his teacher by adding the numbers from 1 to 100 almost instantly.
The child genius had realised that if you group the numbers from 1 to 100 in pairs, the sum is equal to (1 + 100) + (2 + 99) + (3 + 98) + … In other words, 101 + 101 + 101 + … Since there are 50 pairs of numbers, the sum is 101 x 50=5050.
The 100 envelope challenge is a way of saving ￡5050 in 25 weeks. You mark up one hundred envelopes with the amounts from ￡1 to ￡100. Twice a week you choose two envelopes at random, fill each envelope with the amount it says, and put them aside. After 25 weeks, all the envelopes will be full. Bang! More than 5K saved.
Of course, the challenge presupposes that you have ￡5050 to save in the first place and that you possess a large number of coins and banknotes. Still, the tens of thousands of reactions on social media to the trick seem to indicate that many people find it inspiring. “If you have a three-year plan to buy a house, you could have a little over ￡30,000 for a down payment by doing this,” one saver reportedly posted on social media. Perhaps the inclusion of randomness is what makes the exercise more fun.
Today’s puzzles require you to summon your inner Gauss and find fast ways of counting number lists. You are not allowed to use a calculator nor add them all up individually!
1. The double bill
Which of the two additions below (the sum of the nine numbers on the left, and the sum of the nine on the right) adds to the highest number?
2. The 2D envelope challenge
Sum all the numbers in the grid below.
Please NO SPOILERS
I’ll be back at 5pm UK with the solutions.
UPDATE: To read the solutions click here.
While you are here, can I go off topic briefly and ask you to spread the word about the Young Sportswriter of the Year award, run by the Guardian and Football School, the kids book series I co-write with Ben Lyttleton. Children between 7-12 need to submit up to 600 words (before August) about a sportsperson or sports event and can win fantastic prizes, such as their story published by the Guardian.
A fantastic line up of judges includes Tour de France cyclist Tao Geoghegan Hart, Olympic gold medallist Christine Ohuruogu, golfer Ian Poulter and German World Cup winner Per Mertesacker. For more information please click here. With the Euros, Wimbledon, the Tour de France, and imminently the Olympics, there is no shortage of subject matter!
I’m the author of several books of maths and puzzles and also of the book series Football School. The latest book in the series, Terrific Teams, is out this week. It contains stories about 50 amazing teams from Bungay Town to Brazil, and from Lewes to Liverpool.
Sources for today’s puzzles: 1) The Moscow Puzzles by Boris A Kordemsky, 2) Algorithmic Puzzles by Anany and Maria Levitin.