– What’s that…? – Mathina asked, but Leo remained speechless as he stared at the plasma circles. She pushed him gently on the back to wake him from his amazement. He rushed to the entrance of the building.
– “Lo-gi-ci-ty Sta-di-um,” – Leo syllabified the label. Mathina sneaked closer to the building’s wall but saw no entrance, only heard people quarreling inside.
– Sir, you can’t do that to us, – the sound came from the hallway. – Simply give us our prize and let’s go home!
A man in a training suit crossed the wall to the street close to Mathina. A whistle hung from his neck.
Soon, six young men arrived from the wall, too, carrying strange sports devices, Möbius strip-shaped rackets in their hands.
– You can’t simply leave like this! – they begged.
– Alright, I give you another chance. Now you can guess!
The guys looked at each other, tried to count on their fingers, uttered some numbers.
– You can’t figure this out! – two guys complained.
– What is it that you can’t find out? – Mathina asked enthusiastically, but Leo put his hand on her mouth because he could see the others staring at her.
– All right, if this little girl finds the solution for you, you’ll get the gold medal! – the trainer said, and Leo kept his hand on Mathina’s mouth, as he knew she would make a scene for being called “this little girl”.
The trainer explained the situation to Mathina:
– We are Logi-city’s local team and tonight we won the regional logi-soccer championship. But I’m convinced that if we want to perform well in the national competition, it makes sense to train our brains all the time. So: we received a number of gold medals today. And I told them how I’ll distribute these golds. We proceed according to the name list. The first person on the list receives the sum of the digits of the number of golds received. For example, if we got 721 golds, then the first one gets 7 + 2 + 1, i.e. 10 golds. There remains 721 minus 10, or 711 golds. The next one gets the sum of the digits of the gold left this way. And so on. I told them that after two full rounds, we’ll run out of golds, and only George will get more than the others, all the others get identical amounts. Well, knowing this, it takes maybe a second to figure out how many golds we have and who gets how many, but these 6 have been thinking for hours already.
– Where is George on the list? – Mathina asked.
– Little lady, we won’t tell you that. Try to figure it out! – the trainer replied.
– Do you have pen and paper? – Mathina asked.
The trainer produced his pen and notebook and tore out a sheet.
– Thank you! So it has to be split between 7 people, right? – The boys nodded. – Between the 6 team members and the trainer. Mathina started scribbling and cried out after a few minutes – Yeah!
– So how many do we get? How many? – The athletes inquired.
Question: Can you find out how many?
– I’m not going to simply tell you that, – Mathina grinned. She whispered something in the trainer’s ear. The trainer nodded contentedly. – I’d rather help you figure it out, – Mathina said.
– But I know, I know how to solve this! – Leo jumped up. – Let me tell you! Please!
Seeing how excited he was, Mathina and the trainer gave him the word.
– Well, I think we could go through all the cases, like what if all get 1 or 2 or 3 golds, except for George, and so on! And we’ll come across the right number. It won’t be a high number, I am sure…
Leo began to count, thinking out loud.
– If all take always the same amount, except for George, then two consecutive ones will get the same amount. They cannot take 1, because there are no two neighbouring numbers with 1 as a sum of the digits. If each of them takes 2, then it doesn’t work either, since in case the sum of the digits is 2, then the number less or more by two won’t have 2 as the sum of the digits – Leo slowed down and swiped off the sweat from his forehead. – What if we take 3? The sum of the digits is 3 with both 12, 21 and 30, but not with the following player. There is a problem with number 4, too. Is there a solution…?
Leo felt dizzy, although he only considered 4 cases. Mathina looked at him with a questioning look.
– What if we made a try with my method as well?
– For sure! – answered the boys relieved. Mathina couldn’t suppress a triumphant smile as she turned to them.
– Draw this table and make a try with, say, 116 golds.
Question: Do you notice any regularity with 116?
– 116 isn’t good because we’ll run out of it soon, – George said, turning to Mathina.
– Yes, that’s true. But that’s just an experiment for now. Maybe we can learn something from it.
– I may have done something wrong, but I almost always arrive at 9 golds, – George added.
– Yes, you are right! And we can also see an 18…
– But that’s also divisible by 9, – George interrupted quickly.
– Yes! You can, if you want! – Mathina smiled. – And why do you think it is like this?
Question: Why do you think it’s like that, that if we subtract the sum of its digits from a number, we always get a number divisible by 9? Can you prove this in general for three-digit numbers?
Try it on a few other numbers!
– I know! – George shouted. – Let x be the first digit, y the second and z the third. This means that we can put this number down like this: 100x + 10y + z. And the sum of its digits is x + y + z. What happens if we subtract one from the other? 100x + 10y + z – (x + y + z) = 99x + 9y = 9 (11x + y). This way it’s divisible by 9. And this is true for any number of digits, because the last digit is always dropped, and the rest always gets 9, 99, 999,…. 99999999 as a multiplier, because we subtract one from a power of ten.
The trainer raised his head and listened to the argument with bright eyes. He had always talked down to George, but now he saw that he could figure out quite a few good things with a little guidance.
– Good, but we still don’t know how much we’ll get! – the others grumbled.
– Not really, but we’re making progress – Mathina said.
– Listen – George took over again, – we saw in the example that we would get only 9 except for the first one in the line. So I think the first one will get more and the others will all get 2 times 9!
Mathina listened to George with eyes half-closed, and asked him to doublecheck his calculation.
– Good, – George went on. – Then six of us will get 2 times 9, the first one will have 9, that’s 13 times 9, totaling to 117, and the first in the first round will get more, so you need a number greater than 117, a number which has a larger sum of digits than 9. Let’s say 127.
Question: Do you think George is right, does it work with 127?
– George, that’s not good! On the one hand, we’ll run out of gold sooner, and on the other hand, the first and the fourth get a different amount, and the seventh as well, because he gets zero in the last round! – the others told George.
– I think there is a mistake at the fourth, – Leo interrupted.
– Shh, I didn’t ask you, – Mathina shook her head, but then she smiled proudly at Leo.
– The boy is right, – George said. – I forgot I had to give 18 at 99. – Then I’ll modify it like this: the first one must also get 9 in both rounds!
Question: Can you figure out how many golds are actually needed?
– So I thought 6 people would get 9 in both rounds, and one person would get 18 in the first, then 9 in the second round. This means: 7⋅18 + 9 = 135. And then the fifth person gets more than the others!
– But George, we knew you were the fifth on the list, – the others laughed.
– Oh really! – George shook his head. – But you didn’t say so either! – They all laughed. – But it’s okay, this way it feels even better to know the solution!
The trainer passed out the golds one after the other. He gave Mathina a round of applause:
– You’re amazing! You’re welcome to be an assistant trainer of my logi-soccer team, whenever you want! We’re happy to see people like you on the team!”
Mathina thanked him for the opportunity and continued her explorations with Leo. There were new adventures ahead of them.