Solve this puzzle and submit a written solution and ... · difficult puzzles: Given a dozen gold...

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AGMath.comSolve this puzzle and submit a written solution andEXPLANATION to get your name listed below:

Counterfeit Coins

A king has 10 stacks of 10 coins.9 of these stacks are counterfeit, andonly 1 of the stacks is real gold.The fake coins weigh 100 grams each,while the real gold coins weigh 101grams. The king offers this challenge:“Find the stack of gold coins with onlyone weighing, and I will give it to you.”

How can you find the real stack using ascale only once?

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AGMath.comStarting at the top left corner of a stan-dard chess board, a rook moves (nodiagonal moves) across every squareexactly once before ending at the lowerright corner of the board.

Is this possible?

Show how it canbe done, or find away to prove thatit is impossible.

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Solve this puzzle and submit a written solution andEXPLANATION to get your name listed below:

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The Rook’s Corner

end

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Mislabeled Drawers

Scott has three labeled dresser drawersfull of socks. One drawer has all blacksocks. Another drawer has all whitesocks. A third drawer has both black andwhite socks. Unfortunately, all of thedrawers are mislabeled. Scott chooses adrawer and reaches in without lookingand pulls out a white sock. He is able tocorrectly label all of the drawers.

How is this possible?

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The Old Man’s Walk

An old man walks from his house onemile due south, turns and walks one miledue east, turns again and walks one miledue north.

Strangely, the old man finds himself athome again.

What is the old man’s name?

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Pencils

It is possible to place 4 golf balls sothat each ball touches the other three.

Is it possible to place five pencils sothat each pencil touches the otherfour? How about six?

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Simple Geometry

What is the length of segment AB?

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A

B10in

3in 2in

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Toothpick Geometry

Twelve one-inch toothpicks are used be-low to create figures with areas of 9in2

and 5in2. Is it possible to arrange 12toothpicks (all 12 must be used, nobreaking toothpicks) to form a figurewith an area of EXACTLY 4in2?

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Dodgeball

Alvin, Bill, and Carl invent the followingdodgeball game:All three players stand in a triangle andtake turns making one throw. If a playerhits his opponent, that person is elimi-nated. They play until only one player re-mains who is the winner.Al can hit his opponent 50% of the time.Bill can hit his opponent 75% of the time.Carl can hit his opponent 100%.If all three players use perfect strategy,and they go in alphabetical order, what isAl’s best strategy? Who will usually win?

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A cute triangle dissection.

Is it possible to divide an obtuse tri-angle like the one below into severalsmaller acute triangles? If it can bedone, what is the fewest number ofacute triangles that can be created bydividing the obtuse triangle?Below is an example of a failed attempt.

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acute

acuteacute

obtuse

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Windy Flight

A pilot regularly makes round-trip flightsfrom Raleigh to New York City and back.He always flies at the same air speed,and in a straight line. The flight is 500miles each way. With no wind, theround-trip (flight only) takes two hourseach way. With a 50 mph wind blowingdirectly from New York to Raleigh, willthe round-trip flight take more time,less time, or exactly the same amountof time? How long will it take?

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Blue-Eyes

Whenever you meet two of the Jonessisters (assuming the two you meet arealways a random pair), there is a 50%chance that they will both have blueeyes. If three of the Jones sisters haveblue eyes, how many Jones sisters arethere altogether?

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Mountain Climber

A climber leaves at sunrise and ascendsthe narrow trail that leads to the summitof a tall Appalachian peak. He varies hisspeed and takes several rests along theway, finally reaching the peak.The next morning at sunrise he begins towalk down the mountain along the sametrail. Part of the way down, he reaches aspot at the exact same time that he wasthere the previous day on his way up.When he tells his wife, she says “Ofcourse, that was bound to happen.”She offers a simple proof. What was it?

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Chain Links

A woman wants to create a necklace fromseveral short pieces of silver chain. Sinceit takes some effort to cut the chain linksand reconnect them, she wants to makeas few cuts as possible. What is theminimum number of links she will need tocut to connect the necklace as shown?

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Fishy Cut

How can you cut the figure below into twoidentical pieces?

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34

56

78


How Many Triangles?

Below is a diagram of how eight trianglescan be formed with three straight linesover the letter M. What is the maximumnumber of triangles that can be createdusing only three straight lines?

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Easy Area

Find the area of the dashedrectangle below.Explain your reasoning.

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10

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A Counterfeit Coin #1

You are given four quarters, and toldthat one of the quarters is counterfeit.It does not weigh the same as a realcoin, but you are unsure whether it isheavier or lighter. Using a balancescale only twice, can you determinewhich is the counterfeit coin, andwhether it is heavier or lighter than areal coin?(Hint: You also have apocket full of real quartersthat you may or may notneed to use.)

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A Counterfeit Coin #2

This is one of my favorite and mostdifficult puzzles:Given a dozen gold coins, you are toldthat one coin is counterfeit. Thecounterfeit coin does not weight thesame as the other coins. You do notknow if it is heavier or lighter. Using abalance just three times, it is possibleto find the counterfeit coinand determine whether it isheavier or lighter than thereal coins. How can this bedone?

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No! Pie Are Round

Carl has a very strange method ofslicing pie. He tries to make as manypieces as possible with as few cuts aspossible. With three cuts, he hasfound a way to make seven slices.How many slices can he make with sixstraight cuts?

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1 2

54

67

3

2r

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Cut the Cake

Carl made a new discovery when cuttingcake. With just three slices, he was ableto create eight pieces of cake. How isthis possible?

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Paying the Rent

Roger cannot pay his rent for May incash. To pay his rent, he offers hislandlady one inch of a 31-inch gold barhe owns every day of the month. Sincecutting the bar is a lot of work, he de-cides to give her a one-inch piece foreach of the first four days, then on thefifth day he gives her a 5” piece andtakes back the four 1” pieces. He cuts a10” piece and a 12” piece, and is able topay every day’s rent. Is there a betterway he can do this by cutting evenfewer pieces? How?

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UggaMugga!

On a small island, you are studying twotribes: one of truth-tellers and one of li-ars. You come across two natives and askthe skinny one if he is a truth-teller.“UggaMugga!” he replies.You don’t understand,so you ask the fat nativewhat the other native said.“He say ‘yes’, but

him liar!”

Which tribe does each native belong to?

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Tribes

On a small island, you are studyingtwo tribes: one of truth-tellers andone of liars. You come across threenatives: Ak, Bak, and Tak. Ak pointsto Bak and Tak and says to you,”Them two in same tribe.”You ask Tak if Ak and Bak are in thesame tribe. What does he say?

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Orchard

A man is planting an orchard and wantsto keep his rows even, so he buysenough trees to plant eight equal rows.Unfortunately, one dies. He still hasenough trees to plant nine equal rows.Before he is able to plant his trees, one isstolen. Incredibly, the man still hasenough trees to plant ten equal rows.What is the fewest number of trees theman could have purchased initially?

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Handshakes

Five married couples sit down toplay poker (10 people). Beforethey play, each person shakes thehand of everyone at the table thatthey have not met. After theyshake hands, John asks everyoneelse how many hands they shook.The nine other players respond:8, 7, 6, 5, 4, 3, 2, 1, and 0.How many hands didJohn shake? Howmany hands didJohn’s wife shake?

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Short Circuit

You need to connect wires betweeneach of the boxes below from the nodesindicated. Is it possible to connect A toA, B to B, C to C and D to D withoutcrossing wires (or leaving the box)? Ifnot, prove it.

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A B C D

C

B

D A

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In the Kitchen

A father and daughter are in the kitchenmaking dinner when the little girl startsthe following conversation:“Is 60 more than 30?”“Yes honey, 60 is more than 30.”“Is 100 more than 60?”“No honey, 100 and 60 are the same.”“Is 100 more than 90?”“No, 90 is more than 100.”If the father was always correct, whatwere they doing?

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Oh, Brother!

You are buying gifts for your family.You know that your uncle Bobby hastwo children, and you remember meet-ing one of them, who is a boy. Unfortu-nately, you cannot remember whetherthe other child is a boy or a girl. Whatis the probability that Bobby’s otherchild is also a boy?Hint: It is a puzzle.

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Lost Hat

Upstream or downstream, Ben alwayspaddles his canoe twice as fast as the cur-rent he is paddling in (if the current isflowing 2mph, he paddles 4mph relative tothe water). One day, he is paddling up-stream in a river when a gust of windblows his hat off of the back of the boat.Not noticing, he paddles for another 40minutes upstream before turning his boataround. How long will it takefor him to catch-up to hishat in the flowing river ifhe paddles downstream?

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Chicken Nuggets

Chicken nuggets are sold in packs of 6,9, and 20. What is the largest wholenumber of nuggets that you cannot buy?

(Bad Example: Youcan buy 79 nuggetsby purchasing twopacks of 20, threepacks of 9, and twopacks of 6.)

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Fuses

You have two fuses (long strings likethe ones used to light dynamite). Eachof the fuses is a different length, but isguaranteed to burn for exactly onehour when you light the end. Unfortu-nately, the fuses do not burn at a con-stant rate. One fuse may burn 5 inchesin 5 minutes, then a whole foot thenext minute. You need to measure 45minutes with only a lighter and the twofuses. How can you do this?

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Another Orchard Problem

A man is planting apple trees in hisorchard. When he comes in for theevening, his daughter asks him howmany rows he planted.“I planted five rows.”“And how many trees in each row?”“Four trees in each row.”“So you planted 20 trees!” exclaimsthe daughter, anxious to show off hermath ability.“No, I only planted ten trees.”How is this possible?

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Dangerous Overdose

In order to survive a terminal illness, aman takes medication. He has twomedications. If he does not take one(and only one) of each pill, he will die.Unfortunately, the pills look exactly thesame and there is no way to distinguishone from the other. One morning, hespills one pill from bottle A and two pillsfrom bottle B and gets them completelymixed up. How can he make sure hegets the right dosage (one of each pill)without throwing the mixed-up pillsaway and starting all over?

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Calendar Cubes

For your teacher, you create a calendar.First, you make three blocks which showall twelve months. You need to showthe day of the month on two cubes.You need to use the six faces of eachcube to represent each day of themonth, from 01 to 31. How should eachcube be labeled so that every date canbe shown?

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0 6June

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Water Measurement

You have a nine-ounce glass, a four-ounce glass, and a faucet with an end-less supply of water. You can fill ordump either glass. How can you mea-sure exactly six ounces of water? Whatis the smallest amount of water you candump in the process?(note: You are not allowed to makemarks on either glass.)

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4oz.9oz.

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House Number 12

You want to put some fancy letters onyour house to spell-out your address.The letters don’t have price tags.Fortunately the three people in frontof you are also buying letters.The first buys O N E for $2.The second buys T W O for $3.The third buys E L E V E N for $5.How much will it cost to buy theletters to spell out T W E L V E ?Correct explanation necessary, noguess-and-check.

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T W E L V E

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On Again, Off Again

In the school office after hours, youfind a strip of 25 light switches corre-sponding to rooms 1 through 25.All the switches are in the on positionwhen you start flipping them. Youstart by flipping every 2nd switch(2,4,6...) then every third switch(3,6,9...) followed by every fourthswitch (4,8,12...) then every fifth, etc.until you flip only the 25th switch.You check the lights in every class-room. Which rooms have the lightson?

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Bridge Crossing

Five men are crossing a rickety oldbridge in the dark, strong enough tohold only two men. They have onlyone flashlight. Each man travels at adifferent speed. Al takes one minuteto cross, Bill takes two, Carl takes 5,Dan takes 7, and Earl takes 10. Trav-eling together, twomen can only go asfast as the slowerman. What is theshortest amount oftime all five men can take to reachthe other side of the bridge?

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Men With Hats

Four men are arrested and broughtto the local jail. There is not enoughroom for them, so the sheriff makes adeal. Three of the men are lined upfacing forward so that each can onlylook ahead. The fourth is placed in acell out of view. Each is given one offour hats, two white and two striped.If any of the men can guess the colorof his hat (without communicating),all four will be set free, but a wrongguess gets all four men killed. Whatis the probability that at least oneman will know the color of his hat?

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AGMath.comSolve:FOR THIS PUZZLE, SHOW ALL CONSTRUCTION MARKS.

Hard-to-Halve

Draw ONE STRAIGHT LINE which di-vides BOTH of the rectangles belowinto halves of equal area, creatingtwo grey areas of equal area:

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AGMath.comI Like Coconuts

Stranded on an island, Al, Bob, and Carl find abig pile of coconuts. They decide to wait untilmorning to divide them, but in the middle ofthe night Al wakes up and decides to take hisshare. He divides the coconuts into threeequal piles, hides his, and throws one leftovercoconut to a nearby monkey. Bob awakes anddoes the same, followed by Carl, each dividingthe remaining pile, hiding his share andthrowing one leftover coconut to a monkey.In the morning, they divide what is leftequally and there is once again a leftover co-conut (which they throw to a nearby monkey).What is the fewest number of coconuts thatthey could have started with?


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AGMath.comCUBED!

A cube can be subdivided into smaller cubesas shown below. Those smaller cubes can besubdivided as well, creating infinite possibili-ties for dividing a cube into ‘n’ smaller but notnecessarily identical cubes.

Show how a cube can be subdivided into 49smaller cubes.


In this example, thereare 8 small cubes and26 ‘large’ cubes = 34.

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AGMath.comI got the digits!

The sum of 1+2+3+4+5+6+7+8+9=45.The product of these digits is 9!=362,880.

Find nine single digit integers with the samesum and product as the digits 1 through 9.

Bad example:9

x 8

x 8

x 7

x 5

x 3

x 3

x 2

x 1=362,880

but9+8+8+7+5+3+3+2+1=46 (close, but not correct)


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AGMath.comEscape the Nerds!

You have been kidnapped by a band ofdisgruntled math pranksters from MIT!The kidnappers blindfold you, place you into adark room and force you to wear thick gloves.They toss 100 quarters into the room on thefloor and offer you the following deal:“21 of the coins on the floor show heads, theother 79 coins show tails. If you can sort thecoins into two piles, each containing an equalnumber of coins showing heads, we willrelease you.”

How do you escape the nerdy miscreants?


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Date post:	17-Apr-2018
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