Pailyn's Palindromic Pairs: PMQ41

Alice had a fever. She was in bed, her head felt like a cushion and she was tired of being tired. She tried to read but soon drifted off into sleep. But it was not a restful sleep; her mind had wandered into a landscape of numbers.

She saw trains of numbers racing round an imaginary roller-coaster. She saw snakes of numbers biting their tails and rolling away like monocycles on a mission. The whole world seemed like a sea of numbers, some grew bright and shiny as if demanding attention while others faded away into the background. If they were trying to show her some deep and meaningful patterns they never kept still long enough for Alice to figure them out. She somehow felt like they were mocking her like spritely flighty things that were always out of reach.


At one point she saw a line of huge numbers, as tall as a house and as a bright as a box of crayons. They seemed to be weaving their way towards the horizon; Alice followed them. At the end of the line (or should that be the beginning) stood an enormous mirror. Alice got as close as she dared and saw that the numbers were reflected in the mirror; nothing peculiar about that, she thought. But as the line of numbers approached the mirror, each number collided with its reflection and disappeared. Alice stared at this vanishing trick for a while, thinking of what to do next. She got closer to the mirror so that she might see her own reflection – but all she saw was the line of reflected numbers. Suddenly, she felt a push from behind and flew into the mirror – and woke up! Annoying dream, she thought.

Alice felt sticky and sweaty and wasn’t sure if she was feeling too hot or too cold. She was, however, sure that she was thirsty and reached out for a glass of water on her bedside table. Lying next to the glass was a purple envelope with fancy black writing. As she focussed her eyes, she read ‘To Alice’. She opened the envelope and found a crash of colours. It was a card, but not a boring old get-well-soon card, and inside it didn’t have a boring old get-well-soon message – it was a puzzle!

“Here is a perfectly simple product,
ABCDE x N = EDCBA
The number ABCDE is made from 5 different digits.
When it is multiplied by N, which is any number from 1 to 9, ABCDE is reversed to yield EDCBA.
For which values of N can this really happen?
For each value of N also find the number, or numbers, ABCDE.
By the time you finish this, you should be as bright as a sunflower!
From Prof. Pailyn”

Alice started to feel like her normal self again.

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