# Maths puzzle: Can you help this amnesiac get her memory back?

19 Sep 2016

Daphne struggling to remember why all of her friends’ names run in alphabetical order. Image: Shutterstock

What would you do if you were struck down with crippling amnesia? What if you couldn’t form new memories? Would you use trees to help you? What about maths?

She stirred in the bed, opened her bleary eyes and looked around. She had no idea where she was. More troublingly, she had no idea who she was. All she could remember was a name – Daphne.

As everything around her came into focus, she realised she was in a hospital. Seeing a nurse passing the door of her room, she called out.

“Oh my god! Doctor,” he bellowed. “Doctor! She’s awake.”

With a flurry of activity, her room was suddenly full of people in scrubs and white coats.

One face separated itself from the crowd.

“I’m Dr Ramoray,” said the face. “We’re glad you’re finally awake. You’ve been here quite some time. There was an accident. You’ve been in a coma for 15 years. Do you know who you are?”

“Daphne?” she said, still not quite sure. “How did I get here?”

“Well, Daphne, you suffered quite a significant brain trauma. You were brought in without any ID. It was only when we released a photo of you to the media that your friends started arriving. They’re going to be so glad you’re awake.”

“Friends? What friends? I don’t remember any of my friends…”

The room went silent.

“We should do some tests,” said Dr Ramoray.

Hours later, after Daphne had been poked and prodded and scanned and questioned, Dr Ramoray materialised in her room.

“I don’t quite know how to tell you this,” she said. “You have… amnesia.”

Dr Ramoray explained that Daphne’s brain injury had caused irreparable damage. For the rest of her life, Daphne would be unable to remember her past. It was also highly likely that she would be unable to make new memories.

## A trick to remember

Amnesia may keep memories away, but, unfortunately, it doesn’t keep old friends away. Daphne started getting visitors.

Every time they left her bedside, she immediately forgot all of the details she’d learned about them.

Dr Ramoray, with whom Daphne was striking up a romance of sorts – as much of a romance as anyone can have when they regularly forget the face of the person they’re in love with – suggested that she start using a trick to remember things.

When it came to remembering her friends and the relationships between them, Dr Ramoray suggested an old-fashioned tree could help.

“Say that Helen, Bertha and Kathy befriended Chloe, who then befriended Anna, who befriended Indy and Julie. Then say that Erin and Grace befriended Fiona, who befriended you, and you also befriended Julie. Then you’d be left with a tree that looked like this,” said Dr Ramoray, drawing out the relationships on a piece of paper.

All of this made sense to Daphne, and it was agreed that this was what she’d do to keep track of her friendships.

But this is the technological age and, to make Daphne’s life easier and her memories more portable, Dr Ramoray decided to design a program that would transfer the trees to Daphne’s phone, giving her a way to electronically store all the relationships she uncovered.

For the software to work, numbers would be assigned to each of the friends – Anna=1, Berta=2, Chloe=3, Daphne=4, etc – and the relationships detailed by the tree would be input as a single string of numbers.

The sample tree that Dr Ramoray had sketched out, for instance, was input as 10_3_1_10_6_4_6_3_1_0_3.

On the friends’ first visit after this plan was given the green light, Daphne asked them to explain how they were all friends. After they explained the group’s history to her, Daphne sketched out the following tree:

Then this, too, was input into the programme. In this case, the number string was 4_8_5_7_7_1_0_4_5_9_1.

## A twist

Although none of them knew it – not the doctors, not even Daphne herself – Daphne had been an engineer before her terrible accident. Maths, logical thinking, and computer programming were all second nature to her.

It was code, not trees, that would save her memories.

Months passed, with the story of how the friends all got to know each other changing with almost every visit. They’d been friends for decades, and the beginnings had gotten all muddled.

Every visit, therefore, created a new tree, and a corresponding string of numbers.

But Dr Ramoray and Daphne were growing closer and closer, and the good doctor was spending every free moment with the woman she loved. She began to notice that the sketched trees that used to lie around the hospital room were disappearing and, in their place, there were just number strings.

One day, she decided to ask Daphne to explain. Picking up a hastily scribbled string of numbers – 10_10_10_7_7_7_3_3_3_11_0 – she showed it to Daphne.

“How does this make sense to you? What set of relationships is this?” she asked, desperate to understand the workings of her true love’s mind.

What was Daphne’s response? What pattern of friendships – what tree – did this string of numbers represent for her?

Scroll down for the solution to this week’s puzzle.

Daphne and Dr Ramoray gaze into each other’s eyes, each finally able to understand the other. Image: bikeridelondon/Shutterstock

## Solution:

Daphne looked deep into Dr Ramoray’s eyes and explained the code to her.

“The first digit in the code tells me the target of the arrow starting from Anna. The second digit tells me the target of the arrow starting from Berta, and so on.

“If a person is at the top of the tree, they will have no target and, therefore, will be represented by a zero.

“As you know, Anna=1, Berta=2, Chloe=3, Daphne=4, Erin=5, Fiona=6, Grace=7, Helen=8, Indy=9, Julie=10 and Kathy=11.

“I don’t need to, but you could work out the tree from there.”

She took the pen from Dr Ramoray’s lab coat and drew the tree that the string represented:

This week’s maths puzzle comes courtesy of Dr Anca Mustata, lecturer in mathematics at University College Cork (UCC), who is actively involved in the Maths Circles initiative, the Mathematics Enrichment programme in UCC and the Irish Mathematical Olympiad.

Kirsty Tobin was careers editor at Silicon Republic

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