Twelve cups stand upside down. Turn over exactly two at a time. Six turns puts them all right way up. Now eleven cups — and no number of turns will ever do it.
Every other challenge sheet here asks for a value. This one asks whether there IS one, and half the time the honest answer is that there is not. Turning two cups over changes the number standing the right way up by two, or by nothing, or by minus two — always an even change. So that count keeps its odd-or-even-ness forever, and an odd number can never be reached from an even one. No amount of trying will show that; only the argument will.
Why this sheet works
Establishing that something cannot happen is a whole category of mathematical thinking, and it is arguably the one that most separates arguing from calculating. A child can search for an hour and learn nothing except that they did not find it. Two lines about parity settle it for good, and settle every question of the same shape at the same time.
Some of these cannot be done. Say which, and say how you know.
1A panel has 17 switches and 1 of them is on. You may throw exactly 4 of them at a time, never more and never fewer.What is the fewest throws that leaves every switch on? Write NEVER if it cannot be done.Why
218 cups stand upside down on a tray. You may turn over exactly 2 of them at a time, never more and never fewer.What is the fewest turns that leaves them all the right way up? Write NEVER if it cannot be done.Why
Possible or neverUp to 20 things
Name: Date:
Some of these cannot be done. Say which, and say how you know.
319 cups stand on a tray and 2 of them are the right way up. You may turn over exactly 4 of them at a time, never more and never fewer.What is the fewest turns that leaves them all the right way up? Write NEVER if it cannot be done.Why
4A panel has 7 switches and 6 of them are on. You may throw exactly 4 of them at a time, never more and never fewer.What is the fewest throws that leaves every switch on? Write NEVER if it cannot be done.Why
Possible or never · Up to 20 thingsAnswer key
1. There are 17 to work with. One throw changes the number switched on by at most 4, and there are 17 − 1 = 16 to gain, so fewer than 4 throws cannot possibly do it. And 4 can, counting the number switched on as you go: 1 → 5 → 9 → 13 → 17.2. There are 18 to work with. One turn changes the number standing the right way up by at most 2, and there are 18 − 0 = 18 to gain, so fewer than 9 turns cannot possibly do it. And 9 can, counting the number standing the right way up as you go: 0 → 2 → 4 → 6 → 8 → 10 → 12 → 14 → 16 → 18.3. Every turn touches 4 of the 19, and whatever state those 4 were in, the number standing the right way up changes by an EVEN amount — 4, or 2, and so on down to −4. So it keeps its odd-or-even-ness forever. It starts at 2 and would have to reach 19, and 19 − 2 = 17 is odd. No number of turns can ever get there.4. Every throw touches 4 of the 7, and whatever state those 4 were in, the number switched on changes by an EVEN amount — 4, or 2, and so on down to −4. So it keeps its odd-or-even-ness forever. It starts at 6 and would have to reach 7, and 7 − 6 = 1 is odd. No number of throws can ever get there.
