# Solution #4: Hidden depths…

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SOLUTION

Since one of the factors in the product is (x − x), the complete expression becomes
0 × SOMETHING. Therefore, the whole expression is equal to zero.
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CHAMPION PUZZLER

Top marks this week go to Troy R, who was the first to post the correct answer (though obviously I had to delete the post, on the grounds that it was a significant spoiler!):

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SOME THOUGHTS ON USING THIS PUZZLE WITH STUDENTS

This is a neat little puzzle for mathematics learners because it illustrates a few important points:

• Firstly, things that appear complicated can actually simplify to become completely trivial.
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• Secondly, if the path that you are on seems to be heading in a horribly complicated direction, it is often worth looking for an alternative approach. In my experience, with this puzzle, many students begin a painful and futile attempt to multiply out the brackets in sequence.
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• Thirdly, always stop and think about whether you have really considered the full scope of the question you are being asked. Have you thought about every detail, or are there things that you have glossed over? In this question, the key is hiding deep within those three little dots, and is easily overlooked.
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• Finally, if you are really stuck with a problem, try writing it out in as much detail as possible. This is relevant when calculating sums and products, trying to find rules for sequences, etc.; anything where many terms are represented in an abbreviated way.
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In this particular puzzle, the penny may well drop if a student actually starts to write out all twenty-six brackets (and hopefully some time before they have written them all!). While this may seem a long-winded approach (and it is!), it is nonetheless better than just staring at the expression completely stumped.

I have tried this problem with GCSE students (14-16 years old) and A-level students (16-18 years old) and found that it is a nice tool to check on their problem-solving skills alongside their understanding of algebra. It is simple, it invites lateral thinking, it potentially requires persistence and it promotes discussion.

The one thing that I don’t like about the puzzle is that it is not really notationally correct. The alphabet is not a genuine mathematical sequence, so technically speaking the continuation of the sequence within that ellipsis is not properly defined. However, since the meaning is clear, and since “Simplify (x24 − x1) (x24 − x2) (x24 − x3) … (x24 − x26)” clearly would not be as effective a puzzle, I am prepared to put up with this minor irritation. Just.

Anyway, not as challenging as usual, but still a nice question in my opinion. I have a really tough one lined up for Puzzle #5, I promise!
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