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Counting on Letters

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?


Stage: 3 Challenge Level: Challenge Level:1

We had correct solutions to this problem from Toh Pingxuan, Chua Zhu Yu, Natalie Clark and Bethany Doggett, Wen Qiang, Tess Moh, Ming Shu and Shih Ning, Robert Haynes, Yuan Zhang, Michael Brooker, Samantha Butters, Greenian Chiu, Jacqui Eaves and Sandy Emmerson, Chong Ching Tong, Clement Goh Weiming, Chen Wei Jian and Ng Yan Shun, and Charles Heppell.

Lots of you explained your reasoning, and had worked very systematically. One person made a guess at one point, which turned out to be right. However there is sometimes more than one solution to this sort of problem, so it's always worth following up the other possibility to check! Congratulations are due to Freddie Manners, who noticed that in fact you did not need to know what A and D were.

Here is the reasoning that Wen Qiang, Tess Moh, Ming Shu and Shih Ning used.

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Since A is 3, U should be 6 or 7.
Since D is 2, Y should be 1 or 6.

The first and fourth columns must add up to different numbers. There can't be a carry from the second column (A is 3), so there must be a carry from the fifth column to the fourth, so H+H must be at least 10. This means L will be odd.

C must be less than 5 (looking at the first column). It can't be 2 or 3, and it can't be 1, since that would make E be 2. So C is 4, L is 9, E is 8, and the third column tells us that U is 7, since it can't be 2.

We now have:

4 3 7 4 H Y
4 3 7 4 H Y
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8 7 4 9 I 2

The only numbers are left are 1, 5 and 6, and we know that Y is 1 or 6. It is quick to try these and see that the only solution is when Y is 6, H is 5 and I is 1. Therefore CAYLEY is 436986.

Several people had found out about the three mathematicians. Here is a brief summary: if you click on the names, you can read the biographies on the St Andrews University History of Maths website.
Cayley was a 19th century British mathematician, who worked on matrices, and geometry in more than 3 dimensions.
Cauchy was a Frenchman, and Michael Brooker claims that he is best known for having far more theorems named after him than any other mathematician!
Euclid was a Greek, born around 325 BC, who worked on geometry.