Normally I don’t care for combinatorial problems, but this problem from Chalkdust Magazine by Matthew Scroggs seemed to bug me enough to try to solve it. It took me a while to see the proper pattern, and then it was rather satisfying.
“You start at A and are allowed to move either to the right or upwards. How many different routes are there to get from A to B?”
See the Chalkdust Grid Problem
The following is a famous problem of Bachet as recounted by Heinrich Dörrie in his book 100 Great Problems of Elementary Mathematics:
“A merchant had a forty-pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used [in a balance scale] to weigh every integral weight between 1 and 40 pounds [when we are allowed to put a weight in either of the two pans]. What were the weights of the pieces?
(This problem stems from the French mathematician Claude Gaspard Bachet de Méziriac (1581-1638), who solved it in his famous book Problèmes plaisants et délectables qui se font par les nombres, published in 1624.)”
The problem has a nice solution using ternary numbers. See the Weight Problem of Bachet.
(Update 4/10/2019) Continue reading →
This is a fun little problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.
“What is the remainder when the 2008-digit number 222 … 22 is divided by 9?”
(Hint: See The Barrel of Beer) See A Multitude of 2s.
This is an interesting problem from the United Kingdom Mathematics Trust (UKMT) Senior Math Challenge of 2008.
“The length of the hypotenuse of a particular right-angled triangle is given by √(1 + 3 + 5 + … + 23 + 25). The lengths of the other two sides are given by √(1 + 3 + 5 + … + (x – 2) + x) and √ (1 + 3 + 5 + … + (y – 2) + y) where x and y are positive integers. What is the value of x + y?”
See the Right Triangle with Roots.
This turns out to be a fairly challenging driving problem from Longley-Cook.
“Mileage on the Thru-State Turnpike is measured from the Eastern terminal. Driver A enters the turnpike at the Centerville entrance, which is at the 65-mile marker, and drives east. After he has traveled 5 miles and is at the 60-mile marker, he overtakes a man operating a white-line painting machine who is traveling east at 5 miles per hour. At the 35-mile marker he passes his friend B, whose distinctive car he happens to spot, driving west. The time he notes is 12:20 p.m. At the 25-mile marker he passes a grass cutter traveling west at 10 miles per hour. A later learns that B overtook the grass cutter at the 21-mile marker and passed the white-line painter at the 56-mile marker. Assuming A, B, the painter and the grass cutter all travel at constant speeds, at what time did A enter the turnpike?”
See Turnpike Driving.
Here is another Presh Talwalkar problem that seems unsolvable at first glance.
“Every day, a train passes a train station along a straight line track, and the train moves at a constant speed. Two friends, A and B, want to determine how long the train is. Lacking proper equipment, they devise the following method. They first synchronize their walking. Both A and B walk at the same constant speed, and each step they take is the same length. One day they line up back to back at the train station. When the front of the train reaches them, they both start walking in opposite directions. Each person stops exactly as the back of the train passes by. If person A takes 30 steps, and person B takes 45 steps, how long is the train, in terms of steps?”
See the Train Length Puzzle.
This is a great puzzle by H. E. Dudeney involving a very useful technique.
“A man bought an odd lot of wine in barrels and one barrel containing beer. These are shown in the illustration, marked with the number of gallons that each barrel contained. He sold a quantity of the wine to one man and twice the quantity to another, but kept the beer to himself. The puzzle is to point out which barrel contains beer. Can you say which one it is? Of course, the man sold the barrels just as he bought them, without manipulating in any way the contents.”
See the Barrel of Beer.
Futility Closet offers another interesting puzzle:
“A billiard ball is resting on a table that measures 10 feet by 5 feet. A player hits it with no ‘English’ and it strikes four different cushions and returns to its starting point. University of Alberta mathematician Murray Klamkin asks: How far did it travel?”
After solving the problem myself, I verified that Futility Closet provides an answer, but without real justification. So I thought I would write up my solution. See Pool Party.
I was sifting back through some problems posed by Presh Talwalkar on his website Mind Your Decisions, when I found another 3 Jugs problem, which was amenable to the skew billiard table solution from my earlier Three Jugs Problem. Here is his statement:
“A milkman carries a full 12-liter container. He needs to deliver exactly 6 liters to a customer who only has 8-liter and a 5-liter containers. How can he do this? No milk should be wasted: the milkman needs to leave with 6 liters of milk. Can he measure all amounts of milk from 1 to 12 (whole numbers) in some container?”
I also believe I found a case where Talwalkar’s solution to the last question needs revision. See the Three Jugs Problem Redux.
This was a rather intricate puzzle from Presh Talwalkar. I found his solution a bit hard to follow, so I tried for a clearer presentation.
“A servant has a method to steal wine. He removes 3 cups from a barrel of wine and replaces it with 3 cups of water. The next day he wants more wine, so he does the same thing: he removes 3 cups from the same barrel (now with diluted wine) and replaces it with 3 cups of water. The following day he repeats this one more time, so he has drawn 3 times from the same barrel and has poured back 9 cups of water. At this point the barrel is 50% wine and 50% water. How many cups of wine were originally in the barrel? ”
See the Diluted Wine Puzzle.