Maths can be challenging. Mathematical problems can be daunting. Knowing where to begin, and knowing the right method to use, can be hard.
Oftentimes, that challenge can make people give up on the maths before they even try. The Irish Mathematical Trust (IMT) hopes to quash that instinct, fostering a love and a deeper understanding of maths among young children.
We’ve teamed up with the IMT to bring some maths puzzles to our readers, to remind us all that maths can be fun. And yes, just a little bit challenging. But in the best way possible.
This latest instalment from the Irish Mathematical Trust challenges you to apply logic, addition, multiplication and a little lateral thinking.
See if you can solve it.
Find a Penny, pick it up…
Penny, known as Lucky Penny by her friends, is superstitious. For her, the numbers 8, 88, 888, 8,888, etc, are lucky, and 6, 66, 666, 6,666, etc, are all evil.
Furthermore, sums of lucky number (8 + 88 + 888, for example) are lucky, and sums of evil numbers are evil.
Penny considers a number to be particularly lucky if the number is large and can be written as a sum of 8s. Similarly, a number is particularly evil if the number is large and can be written as a sum of 6s.
For instance, Penny thinks 904 is lucky, and not evil, because it is a fairly large number and can be reached using a sum of five 8s.
904 = 888 + 8 + 8
Neither 660 or 666 is lucky, not being reachable using 8s, but both are evil.
Finally, a number can be both lucky and evil. For example, 24 is both lucky and evil as it is made up of both 8s and 6s.
24 = 8 + 8 + 8 and 24 = 6 + 6 + 6 + 6
Powers of 10 are numbers like 10, 100, 1,000, etc. Show that 1,000 and all higher powers of 10 are lucky, but that no power of 10 is evil.
(Hint: How many 8s do we need to get 1,000? 10,000?)
Scroll down for the solution to this week’s puzzle.
This week’s maths puzzle comes courtesy of Prof Stephen Buckley, Head of the Dept of Mathematics and Statistics at Maynooth University. Prof Buckley is actively involved in the Irish Mathematical Olympiad and in mathematics enrichment classes at Maynooth.
The numbers 8, 88, 888, etc., are multiples of 8, which means that they can be obtained as sums of one or more 8s.
8 = 8 x 1
88 = 8 x 11
888 = 8 x 111
Any sums of such numbers will also be multiples of 8. For example:
1,000 = 888 + 88 + 8 + 8 + 8
Here, you add 111 copies of 8, then 11 copies of 8, then one copy of 8, then one other 8, then one other 8. All in all, 125 copies of 8.
Indeed, 1,000 = 8 x125, so 1,000 is a multiple of 8.
The same line of thinking works with any other sums of 8s, 88s, 888s, etc. So a number is lucky whenever it’s a multiple of 8.
By the same logic, but replacing 8 with 6, a number is evil exactly when it’s a multiple of 6.
We can break that down a little further, because 6 is also a multiple of 3. Therefore, any multiple of 3 is also a multiple of 6.
Thus, we can say that for a number to be evil, it must be a multiple of 3. As none of the powers of 10 are multiples of 3 – there is always a remainder of 1 – no such number is evil.
While 10 and 100 are not multiples of 8 – and are therefore not lucky – 1,000 is a multiple of 8, and higher powers of 10 are multiples of 1,000 and, hence, multiples of 8.
Thus, they are all lucky.
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