A Fork in the Road

C

atherine and Heathcliff were Sunday driving in the country. They were looking for a secluded lake they had been told about. As their vehicle crested a short hill, they came to a fork in the road. Catherine stopped the car. “Do you remember,” she asked Heathcliff, “If we were told to go left or right at this fork?”

Heathcliff couldn’t remember the directions. As he thought back, his mind wandered, and he came up with another logic puzzle.

“Cath”, he began. “Imagine there were a couple of guides here and they could give us directions.  Now, suppose one of them always told the truth, but the other one will always told a lie. You can assume that each guide knows whether the other is a liar or a truth teller.”“Finally”, he continued. “Let’s also assume that both guides know which road leads to the lake.

“The puzzle Cath”, he finished, “is to ask a single question of one of the guides and then determine which road to take.

_____________________________

Catherine knew Heathcliff could always distract her like this. She wanted to continue driving, and she was considering flipping a coin to decide which road to take. But, she was a little intrigued by Heathcliff’s puzzle.

She couldn’t just point down a road and ask the first guide “Is this the road to the lake?” There was only a fifty-fifty chance the guide she chose would be the truth teller. So, that approach would be wrong as often as it was right. She wondered if there was a question that each guide would answer the same way. How, she wondered, would they answer the question “Are you a liar?” upon reflection, Catherine realized that both guides would say “No”. She also realized that both guides would say “Yes”, if she asked “Is the other guide a liar?”

Catherine felt that she was onto something, but she wasn’t sure how to proceed. She wondered, “What if I ask them something that I know is true?” “Does 2 plus 2 equal 4?” “That would identify the truth teller,” she reasoned, “But, I only have one question. What if I asked, “Would the other guide tell me that 2 plus 2 equals 4”?  Both guides would answer “No”.  Then, in a flash of inspiration, Catherine had solved the puzzle.

“Ok Heath”, I’ve got it, Catherine said. “I’d ask one guide if the other guide would tell me that the left fork led to the lake.” If the answer was “Yes”, I’d take the right fork and if the answer was “No”, then I’d take the left road.”

“Now” she asked, “Tell me the truth. Which way should we go?”

Counterfeit Coins

C

atherine and Heathcliff were spending a quiet evening at home. Catherine was absorbed in a detective novel and Heathcliff was thumbing his way through a check-out stand magazine.

“Cath,” Heathcliff said, interrupting her reading. “Why don’t you put your book down for a minute? I’ve just solved a puzzle and I’d like to share it with you.” Heathcliff paused while Catherine dog-eared the paperback. “Let me read you the ‘Back Page Puzzler’ from this magazine.”He continued, “It says, imagine you have nine coins. One is slightly heavier than the others. Can you identify the heavy coin, using a two pan balance scale only twice?”

Heathcliff smiled with pride as he explained his solution, “For the first weighing,” he said, “I thought I would put four coins in each pan, leaving one on the table.  But, no matter how I tried, I couldn’t solve it that way.” “So,” He continued, “I decided to put three coins in each pan and leave the remaining three coins on the table. That way, after the first weighing I’d know which group of three contained the heavy coin. Once I knew that, I could put one coin in each pan and leave one on the table. If the scale balanced,” he concluded, “The heavy coin would be on the table. Otherwise, it would be in the lower pan.

“That’s great Heath,” Catherine said, reaching for her novel. “Now, can I get back to my mystery?” “Not just yet Cath,” Heathcliff replied with a grin. “Now, I’ve got a puzzle for you”

Catherine listened as Heathcliff began. “The puzzle I solved was just a warm up for this one” Heathcliff continued. “This time, you have 12 coins. One is counterfeit. It’s either slightly heavier or slightly lighter, than the rest. The challenge is to identify the odd coin using the same kind of two pan balance scale no more than three times.” “Remember,” Heathcliff coached her, “In this puzzle, you don’t know if the odd coin is heavier or lighter than the others.”

Catherine set her book aside and considered Heathcliff’s puzzle. She could already sense this was going to be difficult.

Thinking back to Heathcliff’s earlier puzzle, she decided to begin with 3 even groups of coins. Catherine imagined putting four coins in one pan of the scale, four in the other pan, and the remaining four coins on the table. If the scale balanced, the odd coin would be among those on the table. She paused, “This would be easier if I knew whether the odd coin was heavy or light”. “In that case,” she thought, “I would just put two coins in each pan. Then I’d know which group of two contained the odd coin. Finally, I would weigh those two coins against each other”. “Unfortunately,” she reminded herself, “I don’t know if the coin I’m looking for is heavy or light.”

Catherine thought back to the previous puzzle, “Was there anything she could learn from it,” she wondered. “Well,” she thought, “Heathcliff was able to find an odd coin in a group of three, in one weighing.” “But,” she reflected, “He knew the odd coin was heavy.”  Catherine paused in thought for a moment. Then she got an idea.

“If the scale balanced after the first weighing, the odd coin would be in the group of four coins on the table. After clearing the scale, she would put two of the suspect coins in the pan on one side of the scale and another one of them in the pan on the other side.   Finally, she would add one of the coins she knew was genuine to the second pan.”

“Now,” she thought. If the scale balances the bad coin will be the one I left on the table. And if it doesn’t balance,” she went on, “I still have one weighing to find it.” “Let’s imagine the lower pan of the scale contains the two coins from my group of four suspects” “I’ll know that either one of those coins is heavy or the single suspect coin in the other pan is light”. “I can clear the scale and weigh the two coins from the lower pan against each other. If the scale balances, the odd coin is the light one I took off the scale. If the scale doesn’t balance, the odd coin is in the lower pan.

“Well,” thought Catherine. “That’s reassuring. If the scale balances after the first weighing, I know how to find the counterfeit coin” “But,” She wondered. “What if the scale doesn’t balance?”

Catherine pictured the unbalanced scale. She envisioned four coins in the lower pan and four coins in the upper pan. She wondered, “What would happen if I removed two coins from each pan?” After considering this for a moment, Catherine could see that whether the scale balanced or not, she would be left trying to find the odd coin out of a group of four coins with only one use of the scale remaining. She was pretty sure that was impossible.

“This is a tough puzzle,” Catherine muttered. “I thought it might be.” Heathcliff responded.

Catherine took stock. There were eight suspect coins after the first weighing. She realized the second weighing would have to reduce that number down to three.  To start, she decided to remove three coins from the lower pan. If the scale ended up balancing after the second weighing, she’d be able to find the heavy coin in that group. To replace those three coins, she would put three coins from the table into the lower pan. She did some calculating “Everything would be fine if the scale changed orientation after the second weighing.”  But she realized there would still be too many suspect coins if the same pan was low.  In that case, not only would there be the one remaining coin suspected of being heavy in the low pan. But, there would also be four coins suspected of being light in the higher pan.

Catherine mentally swapped one of the good coins she had put in the lower pan with a suspect coin from the higher pan. However, she quickly realized she would still have four suspect coins, if the scale didn’t change orientation after the second weighing. So, she repeated the procedure and swapped anther coin. This left two good coins and two coins suspected of being light in one pan. The other pan contained one good coin, one coin suspected of being heavy, and two coins suspected of being light.

Catherine slowly grinned. The puzzle was solved. If the scale’s orientation remained unchanged after the second weighing, there would be one coin suspected of being heavy and two coins suspected of being light. She could use the scale a third time to weigh the two coins suspected of being light against each other. This would identify the bad coin.

If the pans reversed orientation, there would be two coins suspected of being light. Finding the light one would be easy. Finally, if the scale balanced, she would know that one of the three coins she removed after the first weighing was heavy. She could find that one easy enough as well.

“I solved your puzzle Heath,” Catherine announced. “You are amazing, Cath!” Heathcliff said, standing up from his chair. “Why don’t I leave you alone for a while now?”

The Counterfeit Coin

The Counterfeit Coin
by Rick Lime

So this week when the rhyme word is hum

And I’ve o’er used ‘cum’,  ‘dum’, ‘thumb’, and ‘bum’

A conundrum I’ll pose

Will you solve it, who knows?

I suspect though, it’s too tough for some

Now the gears in my head start to hum

As the clues for this puzzle forth come

You’ve got 12 coins of gold

But there’s one that is old

It’s weight’s off from the rest, by a crumb

You’ve a scale (not of music to hum)

It’s two pans on a chain, and it’s plumb

With this scale weigh the gold

‘Till at last you behold

The coin others are different from

If at this point you’re all thinking hummm ….

Here’s a clue to begin, don’t be glum

Place some coins in each pan

If they balance you can

Safely say that it’s not in that scrum

Since you now see this isn’t ho-hum

One more thing, please don’t think I’m a bum

The odd coin may be light

Or just overweight, slight

And three weighing’s the goal, good luck chum

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What's on sale?

What's on sale

by Rick Lime

The hardware store patron was thrifty

Can you guess what he bought, that was nifty

One dollar for 4

40’s one dollar more

                                 He paid three for 250 (two hundred and fifty)

Monty Hall's puzzle

Monty Hall's puzzle

By Rick Lime

Monty Hall was a game show host whiz

All acclaim for this puzzle is his

He'd deliver his spiel

Then say "Let's make a deal".

And that show is the scene for this quiz

A young couple was once asked to choose

Curtain A, B, or C, with no clues

Only one holds a prize

So it's no big surprise

There's a two in three chance that they'll lose

Their decision's the one labeled C

And they're happy as kids in a tree

'Cuz the host quickly shows

What he already knows

That there's nothing behind curtain B

But the host has got one final play

And it's argued about to this day

He now says they can switch

And that leads to my pitch

Would you keep C or trade it for A?

Missing Money

Missing Money

by Rick Lime

December 11, 2012

There once were three men from Khartoum

And for thirty pounds they rent a room

So, they each put in ten

But we're not at the end

There's much more to this tale, let's resume

Now, the Inn keeper's erred in the rate

Tells his bellboy  five pounds to rebate

He gives one to each guest 

And then he keeps the rest

Well, this theft makes the landlord irate

Cuz some money is missing, although

Where it possibly went, he don't know

Each paid nine for their kip

Plus the two kept in tip

There's another pound yet, where'd it go?

Who's who at the zoo?

Who's who at the zoo?

by Rick Lime

There were parrots and bears at the zoo

Counting all of their heads, twenty-two

And the group was replete

With some sixty-eight feet

Can you figure out how many flew?

Four Girls

Four Girls

by Rick Lime

Mary's parents had four little girls

Their first, April was spoiled and wore pearls

Daughter two they called May

And then June came their way  

What'd they name their new baby, in curls?

It's a limerick and a puzzle! What have I started?