He Loves Me; He Loves Me Not

Logic Level 4

The pretty Princess of Math was playing "He loves me...he loves me not..." game with her flower. At the start, she would pick 1 petal off and then say "love" alternating with "not love" consecutively for the next picks.

However, instead of picking one at time like in the original game, in the following turns, the number of petals she picked would then run as a series of increasing primes (2, 3, 5, 7, 11...). If there weren't enough petals for the next larger prime, she would restart the new round by picking 1 petal and resume the same method. For example, if there were 10 petals, the number of picked petals would run: 1, 2, 3, 1, 2, 1 (ending with "not love").

Eventually, the Princess finished her game with 1 last petal left, which was also the start of her fourth round, ending with "not love".

If 7 was the largest prime number used in her game, then how many petals did her flower have?

Hint : The Princess suggested you should use logic as well.


The answer is 26.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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2 solutions

José Neto
Oct 23, 2015

First thing we know is that 7 was the highest number that she achieved. So as she was counting: 1, 2, 3, 5, 7, ... After 7, she wouldn't keep going, we know that because 7 was the highest, the the number of petals left must be less than 11. So we're going to try some cases, given that there's a pretty small number of cases.

Now we try and see how would be the counting if it had 10 petals left after the 7: 1, 2, 3, 5, 7, 1, 2, 3, 1, 2, 1 This way, she ends on "love", but we know that she ands on "not love". So it wasn't 10.

Now we try and see how would be the counting if it was 9 petals left after 7: 1, 2, 3, 5, 7, 1, 2, 3, 1, 2. This way it didn't even had 4 rounds. We guessed wrong again.

Now we try and see how the counting would be if it was 8 petals left after 7: 1, 2, 3, 5, 7, 1, 2, 3, 1, 1. This way she ands on "not love" and we have 4 rounds, just as the problem states. So we have a winner.

If we try with 7 petals after 7 and on, we will notice that it is impossible to achieve 4 rounds again, so the answer is unique.

Number of petals: 1+2+3+5+7+8=26

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Actually, if we try 23 petals, the number will run: 1, 2, 3, 5, 7, 1, 2, 1, 1. We can still get the fourth round, but since it ends with "love", your solution is still unique. ;)

Worranat Pakornrat - 5 years, 7 months ago

If 7 is the highest the 1st round is: 1 2 3 5 7 (total 18 ends with love)

After the 1st round the total has to be less then 11, and it has to be 4 rounds:

  • 2nd round: 1
  • 3rd round: 1
  • 4th round: 1

This is the minimum with 4 rounds, but it means it was possible to pick 2 in the 2nd round, so it doesn't work.. The fourth round is fixed. So it's only possible to add numbers on the 2nd and 3rd round. The error is on the 2nd, so let's add a prime on the 2nd round:

  • 2nd round: 1 2
  • 3rd round : 1
  • 4th round : 1

This works numerically but it would end with "love", so not right yet, we have to add petals. If we add on the 2nd round:

  • 2nd round : 1 2 3
  • 3rd round : 1
  • 4th round : 1

It works numerically and it ends with "not love". So we have a solution, total is (18+8=) 26

But was it possible to add on the 3rd round instead? If we do it would go: 1 2; 1 2; 1 wich means it was possible to pick like: 1 2 3; 1 and this is a part of our final solution, so we just add 1 petal for the 4th round and we obtain the same solution.

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