At the end of today's meeting, Andrew has 26 marbles, Louis has 30, Matt has 10, and Rob has 34. Below is a table with each week's totals, starting with the amount everyone had coming into the first week.
There are a lot of clues in this puzzle, and it's very difficult to tell where to begin, since they all seem to build upon one another. In addition, the cumbersome wording makes the problem even more difficult. If all of that wasn't enough frustration, there is the tantalizing comment that last month's totals are read to the group, but that information isn't included for us. (This is given in the hint, in case you didn't look, although it turns out to be unnecessary. As we'll explore below, we can still solve the problem without that information.)
Since there is a lot of information packed into each clue, breaking up the clues into smaller chunks will be very important to solving the problem. It is not necessary to use any mathematical equations to solve the problem; however, they are used in the charts in the solution below to keep things concise. The number of marbles each player has at the end of each of the meetings is represented by a set of variables. For example, the number of marbles Andrew has at the end of the first meeting is represented by A:1.
The last thing to keep in mind before we get started is that a very valuable piece of information comes from the problem's introduction: that there are 100 competition marbles total. Thus, no one person can have more than 100 marbles at a time, and the weekly totals must always add to 100.
In decoding the clues, it would be nice to find a clue (or set of clues) which forms the foundation of the solution: some actual numbers to give us a place to start. That clue turns out to be the first from the second meeting:
- Andrew lost a fifth of his marbles to Rob in his first game, then lost a fifth of his remaining marbles to Louis in his second game.
Why is this a good place to start? If Andrew lost a fifth of his marbles to Rob, that means he has four fifths of his marbles left.
If he then loses one fifth of that amount to Louis, that means he lost one fifth of four fifths, which is four twenty-fifths. Andrew then has sixteen twenty-fifths of his marbles left.
But this sixteen twenty-fifths left at the end must still be a whole number, since you cannot have fractional marbles. This leaves us with only a few possibilities for the number of marbles Andrew could have had at the end of week 1.
Now we’ll turn to the first clue of the first meeting:
- Before losing a third of his marbles to Matt, Louis had 11 more marbles the previous week than what Andrew has today.
This means that Louis had 11 more marbles than what Andrew started with at the second meeting. Thus Andrew could not have had 100 marbles, because then Louis would have had more marbles than were available.
Even more important is the part of the clue telling us that Louis lost a third of his marbles to Matt. This means that the original amount of marbles Louis had was a multiple of three. Now we can take all the possible number of marbles Andrew had, add 11 to each of them, then check to see if they are divisible by 3:
Only one combination works. If Andrew had 25 marbles at the beginning of the second meeting, then Louis had 36 marbles at the beginning of the first meeting, which is divisible by 3. We also know that Louis lost a third of his marbles, which gives us the number of marbles Louis had at the end of the first meeting. (This is where we would pick up the solution if using the hint.)
We also know that Andrew must have had 16 marbles at the end of the second meeting, since he lost a fifth of his marbles against Rob, then a fifth of his remaining marbles against Louis.
The first clue of the third meeting tells us that Matt had twice as many marbles as Andrew had the previous week. Thus Matt had 32 marbles at the end of the third meeting.
The first clue of today’s meeting states that Matt lost a total of 22 marbles. This means that Matt now has only 10 marbles.
The second clue from today’s meeting tells us that Rob has 4 more marbles than Louis, who in turn has 4 more marbles than Andrew. This means that whatever amount Andrew has, Louis has four more than what Andrew has, and Rob has eight more than what Andrew has. Since we know that Matt has 10 marbles and there are 100 marbles total, we can determine how many marbles each person has at the end of today’s game.
This really should be sufficient, since they now know how many marbles everyone had at the end of today, and they could continue playing from there. However, it is possible to determine everyone’s past totals as well, given the clue that Matt remembers that he’s only lost one marble in the last month overall. This means he started the month with 11 marbles.
From the two clues from the first meeting, we know that Matt picked up a third of Louis’ marbles, and lost one to Rob. This means that Matt would have had 22 marbles. Since we know there are 100 marbles total, and we also know how many marbles Andrew and Louis had at the end of the first meeting, we can determine how many marbles Rob had.
The second clue of the first meeting also states that Rob’s total at the end of that meeting matched what Andrew had originally. Since we now know how many marbles Andrew originally had, and already knew how many Louis and Matt had, we can determine how many Rob originally had.
From the second clue of the second meeting, Matt picked up two marbles each from Rob and Louis, increasing his total by four.
This means that Rob and Louis both lost two marbles. But they also picked up some marbles in their games with Andrew. In the first clue of the second meeting, Andrew lost a fifth of his marbles (5) to Rob, then a fifth of the remaining marbles (4) to Louis. (The equations look particularly nasty here, but they just summarize what was said above).
From the second clue of the third meeting, we know that Andrew must have finished that day with 29 marbles, the average of the number of marbles Rob and Louis had the previous week.
We know that Andrew got up to 29 by taking 8 marbles from Rob. Since we now know how many marbles Andrew, Rob, and Matt had that week, we can determine how many marbles Louis had as well.
And that's it. Now we know the weekly totals of everyone's marbles.