Blaine's Puzzle Blog

NPR Sunday Puzzle (Jan 29, 2012): An Equation for 2012

2012-02-02T13:00:00.000-08:00

NPR Sunday Puzzle (Jan 29, 2012): An Equation for 2012:

Q: Write the digits from 1 to 9 in a line. If you put a plus sign after the 2, a times sign after the 4, and plus signs after the 6 and 8, the line shows 12 + 34 x 56 + 78 + 9, which equals 2003. That's nine years off from our current year 2012. This example uses four arithmetic symbols. The object is to use just three of the following arithmetic operations: addition, subtraction, multiplication and division, in a line from 1 to 9 to get 2012 exactly. The operations should be performed in order from left to right. There are no tricks to this puzzle. Can you do it?

I was just about to retire for the evening, but I figured you might need some assistance in solving the puzzle, so your help is... Gee, how do I give you a hint to a math puzzle?

Edit: The hints were "retire" (Social Security Administration = SSA = subtract, subtract, add) and "assistance" and "help" (411 = number of digits to group together, with 3 being assumed for the remaining digits).

A: 1234 - 5 - 6 + 789 = 2012

NPR Sunday Puzzle (Nov 13, 2011): What Comes Next?

2011-11-17T12:00:00.000-08:00

NPR Sunday Puzzle (Nov 13, 2011): What Comes Next?:

Q: What number comes next in the following series: 2, 4, 6, 9, 11, 15, 20, 40, 51*, 55*, 60 and 90?

See, I thought I had the answer to this, but if so, there are a couple numbers missing.

*Update: The consensus seems to be that Henry Hook and Will Shortz overlooked a couple terms in the sequence and it should be 2, 4, 6, 9, 11, 15, 20, 40, 51, 55, 60 and 90. Hopefully everyone is able to solve it now with the corrected wording. If anyone has direct access to Will's email, perhaps they could ask for a similar correction to the puzzle on the NPR website.

Will Shortz has confirmed (see his comment) that he extended Henry Hook's original series (2, 4, 6, 9, 11, 15, 20) and in the process overlooked the numbers above. The NPR website has been updated as well. Thanks to everyone that helped clear this up.

Edit: My hint was "See, I..." which sounds like CI which is 101 in Roman numerals

A: 101 is next in the sequence. When represented as Roman numerals, each number in the series requires exactly two letters (II, IV, VI, IX, XI, XV, XX, XL, LI, LV, LX, XC, CI...)

NPR Sunday Puzzle (Apr 3, 2011): Moby Dick scores an 82

2011-04-07T15:14:00.000-07:00

NPR Sunday Puzzle (Apr 3, 2011): Moby Dick scores an 82:

Q: Assign every letter of the alphabet a numerical value: A=1, B=2, C=3 and so forth. Think of a classic work of literature that has eight letters in its title. When the letters are given a numerical value, they add up to 35. What's the title? Clue: The title has two words.

Clue: 12,672

Edit: First hint was 12 as in Adam-12, second hint was 672 as in the birth year of Venerable Bede

A: ADAM BEDE, the first novel by George Eliot

NPR Sunday Puzzle (Jan. 3, 2010): It All Adds Up to a New Year

2010-01-07T12:02:00.000-08:00

NPR Sunday Puzzle (Jan. 3, 2010): It All Adds Up to a New Year:

Q: Write down the digits from 2 to 7, in order. Add two mathematical symbols to get an expression equaling 2010. What symbols are these?

Yippee! A math puzzle for 2010. The most obvious question is, do you need to get fancy with symbols beyond the standard operations of multiplication, division, addition and subtraction? For example, do you need to use a decimal point, factorials, exponentiation, square roots, etc.? Would Will be so diabolical or would he start us off easy in 2010?

I will say, using just the standard four operations between the digits, you can get 160 different results (4 x 4 x C(5,2) = 160). Of these results, 69 are positive integers. Once you have solved the puzzle for 2010, have fun seeing if you can create any of these results: 1, 623, 1102, 1103, 2291, 4572 or 4573. Also, what's the largest number you can create with just the standard operations?

Edit: The largest number you can form is 2345x6x7 = 98,490. If you study that number you'll see it is 49 times the desired solution of 2010. So just divide by 7 instead of multiplying.

A: 2345 x 6 / 7 = 2010

NPR Sunday Puzzle (Dec. 13): All The Digits Puzzle

2009-12-17T16:07:00.000-08:00

NPR Sunday Puzzle (Dec. 13): All The Digits Puzzle

Q: Name five two-digit numbers that are evenly spaced out — like 32, 34, 36, 38 and 40 — in which all 10 digits from 0 to 9 are used once each. What numbers are these?

The first answer came to me as I was driving and listening to the puzzle. However, as I was typing this clue, I came up with another answer that also works. How to give a clue without giving it away, I don't know.

Edit: The clues were to driving which should have made you think of either a car (ie. Car 54) or golf (ie. 18 holes). Those were the starting numbers of two of the four possible sequences.

The tens digit will either step by 1 (5,6,7,8,9) or by 2 (1,3,5,7,9) always ending on 9. The ones digit will also step by 1 (0,1,2,3,4) or by 2 (0,2,4,6,8) always starting (or ending) on 0. That leads to 4 possible sequences:
Starting with 10 stepping by 22 (20+2)
Starting with 18 stepping by 18 (20-2)
Starting with 50 stepping by 11 (10+1)
Starting with 54 stepping by 9 (10-1)

A: There are four possible sequences of two-digit numbers:
10,32,54,76,98

18,36,54,72,90

50,61,72,83,94

54,63,72,81,90

NPR Sunday Puzzle (Aug 9): Would you like cream and sugar with that?

2009-08-14T07:22:00.000-07:00

NPR Sunday Puzzle (Aug 9): Would you like cream and sugar with that?:

Q: A waitress walks up to a breakfast table with five logicians and asks, 'Does everyone here want coffee?'
The first logician says, 'I don't know.'
The second logician says, 'I don't know.'
The third logician says, 'I don't know.'
The fourth logician says, 'I don't know.'
And the fifth logician says, 'No.'
To whom did the waitress bring coffee — and why?

No doubt the waitress has to be a logician too. This reminds me of a related joke:

René Descartes walks into a diner and orders a cup of coffee. The waitress asks if he wants cream and sugar to which he replies, "I think not". He promptly disappears.

Edit: The answer isn't too difficult if you just think of each person's reply choices...

A: The waitress asks, "Does EVERYONE here want coffee?"

If the first logician didn't want coffee, he knows that not EVERYONE wants coffee so he could have replied "No." But he doesn't yet know what the others are thinking so he answers "I don't know." This tells us he wants coffee.

Similarly, the second logician has determined that the first logician wants coffee. Now we are in the same situation as the first logician. If the 2nd logician didn't want coffee, he could say, "No." But because he replies "I don't know." it means he also wants coffee.

The same logic applies to the 3rd and 4th logicians.

The fifth logican has determined that all the rest want coffee. If he also wanted coffee, he would reply "Yes" because he can accurately determine that everyone wants coffee. However, he doesnt want coffee and therefore answers "No", because he is the lone logician that doesn't want coffee and therefore NOT everyone wants coffee.

Summary:
The waitress (being the consumate logician herself) brings coffee to logicians 1 through 4. Logician 5 is not served coffee.

Friday Fun: Rapidly Rotating Electronic Lock

2009-05-01T00:34:00.000-07:00

It's Friday and you are looking forward to the weekend, but an evil genius has locked you in a room. The door to the room is protected by a special electronic lock with four identical buttons equally spaced along the rim of a circular dial.

Each button toggles an internal switch within the mechanism. You can attempt to open the lock by simultaneously pressing any set of the 4 buttons which will toggle the corresponding switches. If you are lucky enough to thereby align the switches so they are all on or all off, the lock will open. Otherwise the dial begins a spinning cycle that lasts for 1 full minute. When it comes to rest you have no way of knowing which button(s) you pressed previously.

Your captor is returning in 15 minutes. Is there any possible method you can think of that will GUARANTEE that you can open the lock in less than 15 tries? If it is not possible, then let me know why that is the case... so I don't waste my time.

NPR Sunday Puzzle (Apr 26): I Thought NPR Always Did Word Problems

2009-04-30T12:08:00.000-07:00

NPR Sunday Puzzle (Apr 26): I Thought NPR Always Did Word Problems:

Q: If 5=4, 7=5, 8=1 and 26=9, what does 12 equal?

It appears that Will has given us a rare puzzle involving numbers. Have fun figuring it out. I'll give you one clue: 23,041=500

Edit: My first thought was that the answer was the number of letters in the number when spelled out in English. FIVE has 4 letters, SEVEN has 5 letters, TWENTY-SIX has 9 letters. But the puzzle creator deliberately threw us a curve ball with EIGHT which is 1, not 5.

Okay, back to the drawing board. As I hinted in the title, this is still related to words and isn't purely mathematical. If you look closely at the English spelling of each number, you'll see there are ROMAN NUMERALS hidden inside.
fIVe = 4
seVen = 5
eIght = 1
twenty-sIX = 9

A: tweLVe = 55

And spelling out 23,041 in English --> twenty-three thousanD forty-one = 500.

Friday Fun: What's the next number in the sequence?

2009-03-27T17:55:00.000-07:00

Can you figure out the next few terms in the following sequence?

Q: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69...

I'll post the answer some time next week.

New Year's Resolution: Exercise Your Brain

2009-01-02T05:17:00.000-08:00

As long as everyone is making New Year's resolutions, I hope you've made one to get more mental exercise. To start you off, here's a challenging "Cross Number Puzzle." The grid above is filled in like a traditional crossword puzzle, except every answer is a three-digit number (100-999) rather than a word. Warning: some of the clues may have you going in circles but there is a unique solution..

Click here for a printable version of the puzzle. And don't worry, you can get around to your other resolutions, like not procrastinating, and going to the gym later. Go sit on the sofa and work on a puzzle instead!

Across:
1. 3 Down plus 5 Across
3. One-seventh of 8 Across
5. Half of 14 Across
6. A prime number
8. Seven times 3 Across
10. Twice 7 Down
12. A perfect square
14. 9 Down reversed
15. The sum of its own digits, times thirty-seven
16. A perfect square

Down:
1. 13 Down plus 10 Down
2. Average of 9 Down and 14 Across
3. 1 Across minus 5 Across
4. A multiple of three
7. 16 Across minus 1 Across
9. 1 Across plus 5 Across
10. 13 Down plus three hundred
11. 12 Down minus 1 Down
12. Anagram of 4 Down
13. 1 Down minus 10 Down

Friday Fun - Mini-Sudoku Puzzle (nine squares!)

2008-09-26T17:36:00.000-07:00

For all of those that are tired of having to fill in a full Sudoku grid, here's a Mini-Sudoku Puzzle. The goal is to fill the nine squares with just the digits 1 to 9. The only hints provided are the "L-block" hints at each corner. Each value tells you the sum of the five squares that make up the two adjacent edges.

Note: This is not a magic square. You cannot make any assumptions about the totals of the rows, columns or diagonals.

See how quickly you can come up with the unique solution. I'll probably post the answer next Friday. In the meantime, please don't reveal the answer so others can enjoy the puzzle too. Post comments on whether you find this puzzle easy, hard, fun or frustrating. I'd be interested in your solving techniques and times, too.

NPR Sunday Puzzle (Aug 3): Mathematical Synonyms

2008-08-07T12:30:00.000-07:00

NPR Sunday Puzzle (Aug 3): Mathematical Synonyms:

Q: Start with an eight-letter mathematics term. Remove the first, fourth and eighth letters to produce a synonym of the original word. What is it?

A small percentage of the population might struggle on this, but I think most will find this puzzle relatively easy.

Edit: I was probably too obvious with the hints which included obvious synonyms of the answers. Check the comments for other hints that were provided.

A: FRACTION --> RATIO

Friday Fun - Cycling on the Bridge

2008-08-01T06:29:00.000-07:00

Two bicyclists start cycling from opposite ends of a bridge. One cyclist is faster than the other and they meet at a point 2,000 feet from the nearest end. When each cyclist reaches the opposite end of the bridge, he takes a 15 minute rest break and then starts on his on return trip. The cyclists again meet 720 feet from the other end. Assuming each is cycling at a constant speed, how long is the bridge?

Note: There is no mention of the actual speed of each cyclist, or the time that each takes but this problem is solvable. In fact, there is an elegant solution that could be understood by an elementary school student, with basic rules of addition and subtraction. It can also be solved the "hard" way. I'll post the elegant solution next week.

Edit: I've provided an answer in the comments.

Friday Fun - How Long is the Ring Road around Iceland? - Answer

2008-07-25T08:00:00.000-07:00

We should be flying back home from Iceland about this time. Hopefully everyone has had fun with the puzzles while we have been gone. If you haven't had a chance to solve the puzzle about the Iceland Ring Road yet, take a look at last Friday's post and don't read any further. But if you want the answer, read on...

A: Let A be the speed of the first couple and B be the speed of the second couple. After an equivalent amount of time T, one couple has traveled AT miles and the other travels BT miles. For the return, the first couple now travels BT miles in 9 hours, while the other couple travels AT miles in 16 hours.

A = BT/9
B = AT/16

9A = BT
16B = AT

T = 9A/B
T = 16B/A
9A^2 = 16B^2
Take the square root of both sides (which is okay because both are positive)
3A = 4B

This tells us the ratio of their speeds is 4 to 3. In other words, over the same time, the faster couple will travel 4/7 of the ring road, the slower couple will travel 3/7. The difference is 120 miles. And if 1/7 is 120 miles, the whole road is 840 miles.

Friday Fun - How Long is the Ring Road around Iceland?

2008-07-18T08:00:00.000-07:00

My wife and I are taking a leisurely drive around Iceland on the Ring Road... at this point we should be a little more than half way on the East side of Iceland in Egilsstaðir. However, I thought it might be fun to give you a little topical puzzle in honor of our trip.

Q: Two couples leave Reykjavik at exactly the same time traveling opposite directions on the Ring Road around Iceland. When they meet later, one couple has traveled 120 miles farther than the other. After a night's rest in a hotel and some refueling, the couples continue their respective drives. The first couple arrives back at Reykjavik 9 hours later, the second couple takes 16 hours. Assuming that each couple maintains the same constant speed each time they drive, how long is the Ring Road around Iceland?

I'll post the answer next Friday.

How old is Mark?

2008-07-11T00:54:00.000-07:00

For everyone that struggled with the pencil puzzle, here's another algebra puzzle to "stretch your neurons". Pay attention...

The combined ages of Mark and Ann are forty-four years, and Mark is twice as old as Ann was when Mark was half as old as Ann will be when Ann is three times as old as Mark was when Mark was three times as old as Ann.

How old is Mark?

NPR Sunday Puzzle (Jun 29): Anyone have a Pencil?

2008-07-03T12:37:00.000-07:00

NPR Sunday Puzzle (Jun 29): Anyone have a Pencil?:

Q: A man buys 20 pencils for 20 cents and gets three kinds of pencils in return. Some of the pencils cost 4 cents each, some are two for a penny and the rest are four for a penny. How many pencils of each type does the man get?

It's a rare NPR *math* puzzle. Using algebra you could write an equation for the number of pencils, and one for the cost of the pencils. But that results in two equations and three unknowns. Fortunately there are some constraints and a little trial and error will get you the answer. Note: You have to have at least one of each type, so just getting 4 of the first type and 16 of the last type wouldn't work.

Edit: I think the thing that confused most people was they assumed they had to buy 4 of the 1/4 cent pencils, or 2 of the 1/2 cent pencils. You can't make 20 cents with those constraints. Here's how I solved it.

Let A be the number of 4 cent pencils.
Let B be the number of 1/2 cent pencils.
Let C be the number of 1/4 cent pencils.

Number of pencils:
A + B + C = 20 pencils

Cost of pencils:
4A + B/2 + C/4 = 20 cents
Multiplying this second equation by 4 to remove the fractions we have:
16A + 2B + C = 80

Now subtract the first equation to eliminate one variable:
15A + B = 60

There are some obvious constraints on A. Because you need at least one of each type of pencil, none of the values can be 0. That eliminates A = 0 or A = 4. Trying the other values you get:
A = 1, B = 45 --> too many pencils
A = 2, B = 30 --> too many pencils
A = 3, B = 15 --> C = 2

A:
3 pencils (at 4 cents) = 12 cents
15 pencils (at 1/2 cent) = 7 1/2 cents
2 pencils (at 1/4 cent) = 1/2 cent

Catch That Bus!

2008-06-30T02:14:00.000-07:00

The local bus leaves Ashwood at 9:21 am and arrives in Baytree at 12:06 pm on the same day. The express bus leaves Ashwood at 10:00 am, traveling the same route, and arrives in Baytree at 11:40 am. At what time does the express bus pass the local bus if each is traveling at a constant speed?

NPR Sunday Puzzle (Jun 8): 5-Digit Sequence

2008-06-12T12:15:00.000-07:00

NPR Sunday Puzzle (Jun 8): 5-Digit Sequence:

Q: A calculator displays a five-digit number. The first four digits are 8735. These digits form a logical sequence. What is the fifth number in the series?

I was led down the wrong path initially because I didn't read the puzzle carefully. There's an important clue in the question which you'll see if you are bright.

Edit: The key to the puzzle was to realize that these digits were on a calculator. I mentioned that in my clue along with the additional hints of "LED", "see" and "bright". I mentioned the word "segment" in one of my comments too.

A: The next digit is also 5. Each digit is the number of LED/LCD segments that are lit in the prior digit.

Guess this Social Security Number

2008-06-07T15:34:00.000-07:00

A certain Social Security Number has the following qualities:

It uses each of the digits 1 to 9 exactly once (with no zero).
The digits from 1 to 2 (inclusive) add up to 12.
The digits from 2 to 3 (inclusive) add up to 23.
The digits from 3 to 4 (inclusive) add up to 34.
The digits from 4 to 5 (inclusive) add up to 45.
The digit 3 is NOT next to a dash (XXX-XX-XXXX).

What is this unique Social Security Number?

Googol to the Googol-th Power

2008-05-16T17:16:00.000-07:00

At this point everyone should be familiar with the number "googol" which is 10^¹⁰⁰ (10 to the 100th power). Written down it is a one followed by 100 zeroes.

The question this week is:

Q: How many zeroes are there in googol^^googol
(googol to the "googol-th" power)?

Mothers' Day Puzzle for all our Supermoms

2008-05-09T17:05:00.000-07:00

Take the following mathematical equation:

MOM² = AMAZON

Can you replace each letter with a different digit {0 to 9} so that the equation makes sense? The letter will represent that digit everywhere the letter appears.

U.S. Timezone Conundrum

2008-04-25T01:26:00.000-07:00

Wendy lives in a state that is on the West Coast. Edward, on the other hand, lives in a state that is on the East Coast. One day Wendy calls from her home and finds Edward also at home.

"Hey Edward, I'm not so good with timezones. I was wondering. What time is it there?"

Edward, checks his clock and reports back with the accurate time.

"That's funny," says Wendy. "It's exactly the same time here."

Where do Wendy and Edward live and how can this be?

Can you turn 2008 into 73?

2008-04-11T22:33:00.000-07:00

Okay, here's another puzzle in the 2008 series. Can you use the digits in 2008 to form an expression that will equal 73.

If you need the full instructions, check the prior puzzle which had a different target result but the same rules.

2008 Math Expression Puzzle

Hitting the Target Puzzle

2008-04-04T17:02:00.000-07:00

Here's a quick puzzle. In the attached image, a circle is inscribed in a square which is inscribed in another circle.

Of the outside yellow ring, or the inside magenta circle, which has the bigger area, and why?