Derivatives Foundation puzzles, solved step by step
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020One glass holds 100 ml of wine and another holds 100 ml of water. You take a spoonful of wine, tip it into the water and stir. Then you take a spoonful of the mixture and tip it back into the wine glass. Is there now more wine in the water glass, or more water in the wine glass?Prop trading firms
Try it first
Decide before any arithmetic: after the two spoonfuls,
Show the worked solution
Exactly the same. Each glass ends with 100 ml, so whatever wine is missing from the wine glass has been replaced, millilitre for millilitre, by water, and the missing wine can only be in the water glass. With a 10 ml spoon and a thorough stir, the return spoon carries back 0.91 ml of wine and 9.09 ml of water, leaving 9.09 ml of water in the wine and 9.09 ml of wine in the water.
Why does the first spoon feel like it settles the question?
Because it is pure wine going one way and a diluted mixture coming back, so it feels as though more wine travelled. Think instead of two cricket teams of eleven who swap some players and still field eleven each. Each glass ends with exactly 100 ml, so every millilitre of wine that left the wine glass and did not come back has been replaced by a millilitre of water: the two foreign amounts must be equal. The number of team A players now in team B is the number of team B players now in team A, however the swaps were done.
With a 10 ml spoon, the wine glass goes from 100 ml of wine to 90 ml and then back to 100 ml holding 9.09 ml of water, while the water glass goes to 110 ml and back to 100 ml holding 9.09 ml of wine, so the two foreign amounts are equal. What do the millilitres actually look like?
Take a 10 ml spoon. After the first transfer the water glass holds 100 ml of water and 10 ml of wine, 110 ml in all, so a stirred spoonful from it is 10/110 wine. The return spoon carries 0.91 ml of wine and 9.09 ml of water, so 9.09 ml of wine stays behind in the water glass and 9.09 ml of water arrives in the wine glass. The arithmetic confirms the argument, but the argument came first and did not need the spoon size, the stirring or any division.
The relationship10 the spoon, in millilitres 100/110 the share of water in the stirred water glass after the first transfer 10/110 the share of wine in that glass What it says in wordsThe water carried into the wine glass equals the wine left behind in the water glass, both 9.09 ml for a 10 ml spoon.Why do the interviewer's variations not change the answer?
Interviewers vary the story: no stirring, five spoonfuls back and forth, a ladle instead of a spoon. As long as both glasses end at their starting volume, the answer is equal, because the argument uses only the totals. The limitation to say out loud: if the return spoon is a different size from the first, the glasses end at different volumes and the amounts differ, so check the volumes before using the shortcut. The desk lesson is the bookkeeper's: in a closed system, look at the totals before tracking every transfer, the same way a net position check catches a booking error faster than replaying every ticket.
Stage Wine glass Water glass Start 100 wine 100 water After spoon 1 90 wine 100 water + 10 wine After spoon 2 90.91 wine + 9.09 water 90.91 water + 9.09 wine Tracking a 10 ml spoon through both transfers leaves each glass at 100 ml with 9.09 ml of the other liquid, which is what the conservation argument predicted without any arithmetic. Where candidates lose it
The common answer is more wine in the water, because the first spoon was undiluted. It anchors on one transfer and forgets that the second spoon also took some of that wine back.
The second loss is reaching the right answer by long arithmetic and then failing the follow-up, such as an unstirred glass or several transfers, because there was no argument underneath. Give the volume argument first and use the numbers only as a check.
What the interviewer asks next
- The return spoon is 5 ml instead of 10 ml. Which glass now holds more of the other liquid, and by how much?
- You repeat the two-spoon swap many times. What do both glasses converge to?
- Where on a trading desk does checking a total first save you from tracking every transfer?
035There are 100 coins on the table. Players take turns removing 1 to 10 coins, and whoever takes the last coin wins. Do you want to go first, and what is your first move?Quant trading
Try it first
Go first or second, and what is the opening?
Show the worked solution
Go first and take 1, leaving 99. Work backwards: whoever faces 11 coins loses, because any take of 1 to 10 leaves 1 to 10 for the other player to finish. The same holds for 22, 33 and every multiple of 11. From 100, taking 1 leaves 99, a multiple of 11; after that, whatever the opponent takes, you take 11 minus it, stepping down 88, 77, 66 and so on to 0, where you take the last coin.
Why work backwards from the last coin?
If you are climbing stairs with a friend and the rule is that the person who steps onto the top stair wins, you do not plan from the bottom; you ask which stair you must leave your friend on so that they cannot reach the top in one go. Games with a fixed last move are solved from the end: find the positions where the player to move loses, then find the positions from which you can push your opponent onto one of them. With 1 to 10 coins allowed, facing 1 to 10 coins is a win, you take them all. Facing 11 is a loss, because every move leaves between 1 and 10. Facing 12 to 21 is a win, since you can reduce to 11. Facing 22 is a loss again. The losing positions repeat every 11.
Every multiple of 11 from 0 to 99 is a losing position for the player who must move, so the first player takes 1 to leave 99 and then answers every take of t with 11 minus t, stepping down through 88, 77 and 66 until the last coin. How do you find the period without listing every position?
The period is the largest take plus one, 11, because that is the one total a pair of moves can always be made to add up to: whatever your opponent takes between 1 and 10, you can take the balance of 11. The losing positions are the multiples of the largest take plus one, and the winning opening move is the remainder when the pile is divided by that number. 100 divided by 11 is 9 remainder 1, so take 1. If the rule allowed 1 to 7 coins, the period would be 8 and the opening would be 100 mod 8, which is 4. If the pile had been 99 to start with, you would want to go second, because the first player cannot leave a multiple of 11.
The relationship11 the largest allowed take plus one, the amount you can always complete in a pair of moves 100 mod 11 the remainder when 100 is divided by 11; take exactly this many What it says in wordsTake the remainder on your first move, then keep each pair of moves summing to 11.What changes if the last coin loses instead of wins?
Then you want to hand your opponent the last coin, so the position you avoid facing is 1 coin, and the losing positions shift up by one: 1, 12, 23 and so on up to 100. Facing 100 in that version you are already lost, so you would want to go second, which shows the interviewer that you re-derive the pattern rather than remember it. The method is the same in every variant: name the terminal position, step back one move at a time to find the first losing position, then find the period. The limitation of the trick is that it needs a game with perfect information and no chance; add a die that sets each turn's maximum and the clean period disappears.
Where candidates lose it
The common loss is taking 10, because a bigger move feels like a stronger start. It leaves 90, which is not a multiple of 11, and a prepared opponent takes 2 to leave 88 and wins from there.
The second is knowing the answer and not the reason. Say why 11 is the period: any take of 1 to 10 can be completed to 11. Without that sentence the interviewer will change the numbers and watch you stall.
What the interviewer asks next
- Players may take 1 to 7 coins instead. Do you go first, and what is the opening?
- The player who takes the last coin loses. Do you go first?
- There are two piles, 100 and 60, and you may take from either pile. Who wins?
- Each turn a die sets the maximum take. Is there still a strategy, and what is it?