The Bogotá Puzzles

By Bernardo Recamán

Calculate the number of a neighbor's grandchildren based on their ages, figure out the denominations of a government's postage stamps, swap the positions of two chess pieces in the least number of moves, and solve other intriguing puzzles with this captivating compendium.
Inspired by such illustrious collections as the The Canterbury Puzzles, The Moscow Puzzles, and The Tokyo Puzzles, Colombian mathematician and professor Bernardo Recamán assembled these 80 brainteasers, which he acquired while living and working in Bogotá. Recamán's colleagues, students, friends, and acquaintances contributed to this stimulating potpourri of word problems, sudoku-style challenges, and other math-based diversions. Complete solutions appear at the end.

• Language: English
• Publisher: Dover Publications
• Release date: Oct 14, 2020
• ISBN: 9780486848334

Book preview

2019

1. Magical Ellipses

These four ellipses represent four sets and all the possible intersections between them. There are eight regions inside each ellipse and fifteen regions among all of them.

Is it possible to assign the numbers 1 to 15 to the fifteen regions so that the sum of the numbers in each of the four ellipses is the same?

2. My Neighbor’s Grandchildren

I asked my neighbor how many grandchildren he had. This is what he replied:

Well, I don’t exactly know—lots. What I do know is that when I saw them last week, they all had different ages. Curiously, the sum of their ages was precisely my own age—that is, 73. I discovered that no other set of numbers whose sum is 73 has a product greater than the product of my grandchildren’s ages.

How many grandchildren does my neighbor have?

3. Triangle of Primes and Squares

Place a different prime number or perfect square in each of the fifteen circles that make up the triangle below. The number in any circle that lies on two others should be precisely the sum of the numbers in those circles. The number in the apex should be as small as possible.

4. Economical Numbers

A number is said to be economical if its canonical prime factorization uses up fewer digits than the number itself. A number’s canonical prime factorization is the unique way of expressing any whole number as the product of increasing prime powers. For example, the canonical prime factorization of 1776 is 2⁴ x 3 x 37.

Most numbers are not economical. The number 1810—Colombia’s year of independence—is far from being one because its canonical prime factorization (2 x 5 x 181) uses five digits as opposed to the four used by the number itself. The same is the case for 1776.

The smallest economical number is 125, whose canonical prime factorization (53) uses just two digits.

There is only one pair of consecutive numbers less than 5,000 that are both economical.

Can you find it?

5. Mxied Smus

It has been swhon taht to raed a txet the oedrr in wihch the lrtetes of ecah idniadiuvl wrod aepapr is not ipmotanrt, so lnog as the fsrit and lsat ltetres are crorect. Tihs is not the csae wtih nmuebrs baecsue if one slcarbmes the ditgis of a nmbeur, it is not psisolbe to wrok out waht the ogirianl nemubr was.

Tehre are cirtaen cesas in wchih tehre is sifuficnet inmoartfion to fnid out the onriiagl neumbrs if olny the interior diigts of ecah of them wree mxied wlihe the fsirt and lsat digtis wree lfet untaelred. Scuh is the csae in the fonlwloig aditiodn:

Can you rseotre the oigrianl atdiiodn and its sum?

6. Postage Stamps

The postal authorities of Alphagonia wish to print six stamps of different whole number denominations so that at most two stamps are necessary to pay the postage for up to 20 alphas. The alpha is the country’s official currency.

Can you find six values for the stamps of Alphagonia that satisfy the conditions set by its postal authorities?

7. Odd Number

Think of any whole number. Now multiply that number by the next nine whole numbers. You will end up with a very large number terminating with lots of zeros. It is almost certain that the first digit (from right to left) that is not zero is an even digit: 2, 4, 6, or