This is a 2017 Center of Mathematics Problem of the Week.
“The average age of a 10 member committee is the same as it was 4 years ago, because an older aged member has been replaced by a younger aged member. How much younger is the young member than the older member they replaced?”
See the Young and Old Problem.
This is an interesting algebra problem from BL’s Weekly Math Games, which is behind a subscription wall.
(a3 – b3)/(a – b)3 = 73/3,
what is a – b?”
In fact, it is possible to solve for a and b individually as well.
See a Tricky Ratio Puzzle for the solution.
(Updates 11/6/2024, 11/9/2024) Other Solutions Continue reading
This is a fairly straight-forward problem from the 1999 AIME problems.
“The two squares shown share the same center O and have sides of length 1. The length of AB is 43/99 and the area of octagon ABCDEFGH is m/n where m and n are relatively prime positive integers. Find m + n.”
See the Another Octagonal Area Problem for solutions.
Presh Talwalkar presented an interesting puzzle that originated in the Ladies’ Diary of 1739-40, was recast by Henry Dudeney in 1917, and further modified using American money.
“Each of three Dutchmen, named Hendrick, Elas, and Cornelius has a wife. The three wives have names Gurtrün, Katrün, and Anna (but not necessarily matching the husband’s names in that order). All six go to the market to buy hogs.
Each person buys as many hogs as he or she pays dollars for one. (1 hog costs $1, 2 hogs are $2 each, 3 hogs cost $3 each, etc.) In the end, each husband has spent $63 more than his wife. Hendrick buys 23 more hogs than Katrün, and Elas 11 more than Gurtrün. Now, what is the name of each man’s wife?”
See the Three Dutchmen Puzzle for solutions.
This is a rather mind-boggling problem from the 1947 Eureka magazine.
“Six men, A, B, C, D, E, F, of negligible honesty, met on a perfectly rough day, each carrying a light inextensible umbrella. Each man brought his own umbrella, and took away—let us say “borrowed”—another’s. The umbrella borrowed by A belonged to the borrower of B’s umbrella. The owner of the umbrella borrowed by C borrowed the umbrella belonging to the borrower of D’s umbrella. If the borrower of E’s umbrella was not the owner of that borrowed by F, who borrowed A’s umbrella?”
See the Umbrella Problem for solutions.
This is a nice little puzzle from the 2024 Math Calendar.
“Find the sum of the coefficients of
(1 + x + x2)3 “
As before, recall that all the answers are integer days of the month.
See the Simple Polynomial Puzzle for a solution.
Since Twitter (now X) is no longer public, I was afraid I would not have access to new Catriona Agg puzzles, but she has put them up on Instagram, which is partially available to the public. I managed to find a half dozen interesting new brain ticklers.
This is an interesting problem from the collection Five Hundred Mathematical Challenges.
“Problem 251. Let ABCD be a square, F be the midpoint of DC, and E be any point on AB such that AE > EB. Determine N on BC such that DE || FN. Prove that EN is tangent to the inscribed circle of the square.”
See the Parallel Lines Problem.
This is a challenging problem from the c.100AD Chinese mathematical work, Jiǔ zhāng suàn shù (The Nine Chapters on the Mathematical Art) found at the MAA Convergence website.
“Now a good horse and an inferior horse set out from Chang’an to Qi. Qi is 3000 li from Chang’an. The good horse travels 193 li on the first day and daily increases by 13 li; the inferior horse travels 97 li on the first day and daily decreases by ½ li. The good horse reaches Qi first and turns back to meet the inferior horse. Tell: how many days until they meet and how far has each traveled?”
The solution involves common fractions, which the Chinese were already adept at using by 100 BC.
See Horses to Qi for a solution.
This is an interesting problem from 180 BC China.
“In one day, a person can make 30 arrows or fletch [put the feathers on] 20 arrows. How many arrows can this person both make and fletch in a day?”
It turns out the solution to this problem led me into the history of numerator/denominator (aka common) fractions, a subject I had been finding difficult to track down.
See Making Arrows for a solution.
