Puzzles involving squares, cubes and higher powers
 W I N I O N O W
SQUARES and POWERS
The Contest Center
Wappingers Falls, NY 12590
 W I N I O N O W

We will post the names of anyone who solves these puzzles.
The puzzles are ranked according to difficulty
* indicates an easy to moderate puzzle.
** indicates a tough puzzle.
*** indicates a puzzle for expert solvers.
#* indicates the answer is partially known.
### indicates the answer is not known.

* Arithmetic Progression
The squares 1, 25 and 49 are in arithmetic progression, with a common difference of 24. Find all sets of 3 squares in arithmetic progression.

Solved by:   Lee Morgenstern, Paul Cleary, P.M.A. Hakeem, Arthur Vause, Claudio Baiocchi

** Arithmetic Progression #2
The squares 1, 25 and 49 are in arithmetic progression, with a common difference of 24. Show that it is not possible to have 4 squares in arithmetic progression.

Solved by:   Lee Morgenstern

** Triangular Cube (contributed by Naim Uygun)
If N is a natural number, the triangular number T(N)=1+2+3+...+N is given by N(N+1)/2. Find a prime P such that the sum of the divisors of T(P) is a cube.

Solved by:   Paul Cleary, Ender Aktulga, Mark Rickert

* Multiples of Powers (contributed by Paul Cleary)
Find the smallest positive integer which is 2 times a square, 3 times a cube, 5 times a fifth power, 7 times a seventh power, and 11 times an eleventh power. For example, 648 is 2 times the square of 18, and 3 times the cube of 6.

Solved by:   P.M.A. Hakeem, Arthur Vause, Claudio Baiocchi, Ionut-Zaharia Chirila, Ender Aktulga, Mark Rickert

* 4 Squares (contributed by Mark Rickert)
Find 4 squares such that the sum of any 3 of them is also a square.

Solved by:   Lee Morgenstern, P.M.A. Hakeem, Paul Cleary, Arthur Vause, Naim Uygun, Ionut-Zaharia Chirila, Ender Aktulga

* 4 Squares #2 (contributed by Paul Cleary)
There are infinitely many sets of positive integers A < B < C < D such that A2+B2,  A2+B2+C2, and A2+B2+C2+D2 are all squares. Find the value of A <= 1500 which leads to the largest number of solutions.

Solved by:   P.M.A. Hakeem, Arthur Vause

Cubic Diophantine (Contributed by Paul Cleary)
Find 3 sets of positive integer solutions to the equation   A3 + B3 = 19C3.

Solved by:   P.M.A. Hakeem, Naim Uygun, Arthur Vause, Ionut-Zaharia Chirila (4 solutions), Ender Aktulga

Cubic Diophantine #2 (Contributed by Mark Rickert)
Find positive integers A, B, C, D, E, F, G satisfying A3 + B3 = C3 + D3 = E3 + F3 = 19G3.
Please submit primitive solutions only, that is, A, B, C, D, E, F, G should not have a common factor.

Solved by:   Arthur Vause (31 solutions), Paul Cleary, Naim Uygun (2 solutions), Ender Aktulga (2 solutions)

** Three Sums Are Squares
Are there 3 positive integers such that their sum is a square, the sum of their squares is a square, and the sum of their square roots is a square?

Sum of 3 Squares (contributed by Paul Cleary)
The equation a2 + b2 + c2 = d2 has an infinite number of solutions in positive integers, however if we restrict a and b to <= 3600 then there are only a finite set of solutions.
Find the values a <= b <= 3600 which give the largest number of solutions.

Solved by:   P.M.A. Hakeem, Arthur Vause

#* Sum of 3 Squares #2 (with Paul Cleary)
The equation a2 + b2 + c2 = d2 has an infinite number of positive integer solutions. (A) Prove that there are infinitely many solutions for which c is a multiple of ab. (B) Find a general formula that generates all such solutions.

Solved by:   P.M.A. Hakeem (Part A), Arthur Vause (Part A)

** Palindromic Equation (contributed by Paul Cleary)
Show that the equation 987a + 789b + 12321c = (a + b + c)2 has infinitely many integer solutions. [Do not include solutions where a+b+c=0.]

Solved by:   P.M.A. Hakeem, Nick McGrath, Arthur Vause, Ionut-Zaharia Chirila

** Rational Powers
Let x be a rational number x=p/q with q>1 and gcd(p,q)=1. Let a, b and c be positive integers, with ax+bx=cx and gcd(a,b,c)=1. Prove that a, b and c must be q-th powers, or find a counterexample.

Solved by:   Arthur Vause

### Squares on a Cube
It is possible to write a square on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a square. For example, write 1 on two opposite faces, and 4 on the other 4 faces. The sum of the 3 faces surrounding each vertex is then 1+4+4=9, a square.
Can this be done using the squares of 6 distinct whole numbers? If so, what set of 6 such squares has the smallest sum?

** Arrange 1 to N #1
Arrange the integers from 1 to N in an order such that the sum of any two consecutive terms is a square. For what values of N do solutions exist? A solution for N=17 is

16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8, 17

Solved by:   P.M.A. Hakeem, Ionut-Zaharia Chirila

** Arrange 1 to N #2
Arrange the integers from 1 to N in an order such that the sum of any two consecutive terms is a cube. For what values of N do solutions exist?

* Arrange 1 to N #3
Arrange the integers from 1 to N in an order such that the sum of any two consecutive terms is a power of 2. For what values of N do solutions exist?

Solved by:   P.M.A. Hakeem, Arthur Vause, Ionut-Zaharia Chirila, Mark Rickert

* Square Digits
Find all squares S = n2 such that when you add S and its digits the result is also a square. For example, if S were 25 then 25+2+5 would also have to be a square.

Solved by:   Xavier Manach, Ken Duisenberg, Colin Bown, Sudipta Das, Stephane Higueret, Nick McGrath, Hai He, Hareendra Yalamanchili, Hashim Mooppan, Marc Schegerin, Arijit Bhattacharyya, Denis Borris, Mark Rickert, Paul Cleary, Tan Lye Huat, P.M.A. Hakeem, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga

* Largest Power (contributed by Nick McGrath)
5^3 is 125, with 3 digits. 8^5 is 32768 with 5 digits. Find the largest N-digit number which is an N-th power.

Solved by:   Ritwik Chaudhuri, Sudipta Das, Marc Schegerin, Hashim Mooppan, Andreas Abraham, Nanda Appadoo, Denis Borris, Shyam Sunder Gupta, Mark Rickert, P.M.A. Hakeem, Paul Cleary, Claudio Meller, Naim Uygun, Arthur Vause, Ionut-Zaharia Chirila, Ender Aktulga

* 3 Cubes (contributed by Sudipta Das)
The digits of 53, 125, can be rearranged to form 83, 512. Find the smallest cube whose digits can be rearranged to form 2 other cubes.

Solved by:   Nick McGrath, Gaurav Agrawal, Andreas Abraham, Denis Borris, Shyam Sunder Gupta, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga

* 4 Cubes (contributed by Sudipta Das)
The digits of 53, 125, can be rearranged to form 83, 512. Find the smallest cube whose digits can be rearranged to form 3 other cubes.

Solved by:   Nick McGrath, Andreas Abraham, Shyam Sunder Gupta, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Ionut-Zaharia Chirila, Ender Aktulga

* 5 Cubes (contributed by Nick McGrath)
The digits of 53, 125, can be rearranged to form 83, 512. Find the smallest cube whose digits can be rearranged to form 4 other cubes.

Solved by:   Sudipta Das, Andreas Abraham, Shyam Sunder Gupta, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Ionut-Zaharia Chirila, Ender Aktulga

No-Zero Squares
Are there an infinite number of squares that do not contain the digit 0?

Solved by:   Nick McGrath, Carlos Rivera, Andreas Abraham, Shyam Sunder Gupta, Paul Cleary, Arthur Vause, Ionut-Zaharia Chirila

First and Last 20
Find the smallest integer N > 1010 such that the first 20 digits of N2 are the same as the last 20 digits of N2. For example, the first 3 digits of 2773632 = 76930233769 are the same as the last 3 digits. [No brute-force computer searches, please.]

Solved by:   P.M.A. Hakeem, Mark Rickert, Paul Cleary

Power Digits
Let N=d1d2d3...dn be an n-digit decimal number, with n>1. Form the sum
S(N) = d1n + d2n + d3n + ... + dnn
Prove that there are only a finite number of integers N for which S(N)=N. (Extra Credit: find them.)

Solved by:   Nick McGrath (found 18), Denis Borris (found 42), Mark Rickert (found 23), P.M.A. Hakeem (found 20), Paul Cleary (found all solutions), Naim Uygun (found 21), Ionut-Zaharia Chirila (sent 11)

** Repeated Digits
The square of 88 is 7744, where each digit of the square is repeated. Find additional squares (not ending with 0) where every digit is part of a repeated sequence of digits, such as 11000555544. We will list the number of squares found by each solver. The notation +F indicates that the solver also found an infinite family of solutions.

Solved by:   Gerald Harrison(9), Sudipta Das(10), Andreas Abraham (5+F), Bipin Kumar (4), Mark Rickert (4), Paul Cleary (8), P.M.A. Hakeem (6+F), Nick McGrath (4+F), Naim Uygun (4), Arthur Vause (24+2F), Ionut-Zaharia Chirila (1), Ender Aktulga (2)

** Repeated Digits #2
Find 3 squares (not ending with 0) in which all of the digits come in at least pairs, and most of the digits come in triplets or longer sequences. For example, if the number 11000222244 were a square, it would qualify because 7 out of the 11 digits are in triplets (000) or longer sequences (2222).

Solved by:   Sudipta Das, Andreas Abraham, Paul Cleary, P.M.A. Hakeem, Mark Rickert

** Sum of Cubes
It is well known that the number 1729 is the smallest integer which can be expressed in two different ways as the sum of two cubes (of positive integers), namely 13+123 = 1+1728 and 93+103 = 729+1000.
Find the smallest integer that can be expressed in two different ways as the sum of three cubes.
Find the smallest integer that can be expressed in three different ways as the sum of two cubes.

Solved by:   James Layland, Sudipta Das, Nick McGrath, Andreas Abraham, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Claudio Meller, Naim Uygun, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga, Ahmet Saracoglu

** N Different Ways
There is a series of problems to find the smallest integer that can be expressed as the sum of 2 squares N different ways for N=2,3,4,... Is there a limit?

Solved by:   Carlos Rivera, Nick McGrath, Paul Cleary, P.M.A. Hakeem, Ionut-Zaharia Chirila

** Swapping Sides
Prove that there is no integer N such that when you swap the left and right halves of N the left and right halves of N² also swap. For example, if N were 3456, whose square is 11943936, then the square of 5634 would need to be 39361194. None of the 4 numbers involved may have leading zeroes (otherwise trivial solutions like 40²=1600, and 04²=0016 would be possible).

Solved by:   Jean Jacquelin, P.M.A. Hakeem, Arthur Vause

* Doublestring Square (contributed by Nick McGrath)
We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12391239 a doublestring number. Find the smallest doublestring number which is a square.

Solved by:   Sudipta Das, Andreas Abraham, Denis Borris, Shyam Sunder Gupta, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Claudio Meller, Arthur Vause, Oscar Lanzi, Ionut-Zaharia Chirila, Ahmet Saracoglu, Ender Aktulga

** Doublestring Square #2
We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12391239 a doublestring number. Are there an infinite number of doublestring squares?

Solved by:   Nick McGrath, Mark Rickert, P.M.A. Hakeem, Arthur Vause, Paul Cleary

*** Doublestring Cube (contributed by Sudipta Das)
We will call a number that consists of the same sequence of digits repeated twice, such as 11 or 12931293 a doublestring number. Find the smallest doublestring number which is a cube.

### Triplestring Square
We will call a number that consists of the same sequence of digits repeated three times, such as 555 or 705570557055 a triplestring number. Find the smallest triplestring number which is a square.

* Triplets #1
In many squares nearly all of the digits come in triplets. For example, in the square 4712865382 = 222111000900025444 all but the 3 underlined digits 222111000900025444 are in triplets. [A sequence of 3+d identical digits will be considered to be 1 triplet plus d extra digits.]
Prove that there is no square consisting entirely of triplets.

Solved by:   P.M.A. Hakeem

* Triplets #2
In many squares nearly all of the digits come in triplets. For example, in the square 4712865382 = 222111000900025444 all but the 3 underlined digits 222111000900025444 are in triplets. [A sequence of 3+d identical digits will be considered to be 1 triplet plus d extra digits.]
Find the largest square consisting of triplets plus 1 extra digit.

Solved by:   P.M.A. Hakeem

** Triplets #3
In many squares nearly all of the digits come in triplets. For example, in the square 4712865382 = 222111000900025444 all but the 3 underlined digits 222111000900025444 are in triplets. [A sequence of 3+d identical digits will be considered to be 1 triplet plus d extra digits.]
Your choice of (A) find the largest square with only 2 extra digits, or (B) prove that there are an infinite number of such squares, or (C) find a 50-digit square consisting of 16 triplets plus 2 extra digits, and containing all of the digits 0 to 9.

Solved by:   P.M.A. Hakeem

** N and N-Squared (submitted by Sudipta Das)
Find the integer N such that N and N2 together contain all 10 digits from 0 through 9 twice each, and also have the largest number of double digits (such as 11 or 66).

Solved by:   Nick McGrath, Andreas Abraham, Shyam Sunder Gupta, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Oscar Lanzi, Ender Aktulga, Ahmet Saracoglu

*** N and N-Squared #2
Find the smallest integer N such that N and N2 together contain each digit its own number of times, that is, 0 zeros, 1 one, 2 twos, ..., 9 nines.

Solved by:   Jean Jacquelin, Mark Rickert, P.M.A. Hakeem, Arthur Vause, Ahmet Saracoglu, Naim Uygun

** N and N-Cubed (submitted by Sudipta Das)
Find the smallest integer N such that N and N3 together contain all 10 digits from 0 through 9 the same number of times.

Solved by:   Nick McGrath, Andreas Abraham, Paul Cleary, P.M.A. Hakeem, Mark Rickert, Naim Uygun, Arthur Vause, Ender Aktulga, Ahmet Saracoglu

** N and N-Cubed #2 (submitted by Arthur Vause)
Find the smallest integer N such that N and N3 each contain all 10 digits from 0 through 9 the same number of times.

Solved by:   Naim Uygun, P.M.A. Hakeem, Paul Cleary, Mark Rickert, Ender Aktulga, Ahmet Saracoglu

** Square Rearranger
Find the smallest three distinct whole numbers A, B and C such that you can rearrange the digits of A and B to get C2, the digits of A and C to get B2, and the digits of B and C to get A2. [Leading zeroes are not allowed.]

Solved by:   Arthur Vause, Naim Uygun, P.M.A. Hakeem, Mark Rickert, Paul Cleary

* Cube Rearranger (contributed by Sudipta Das)
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M3 to get N3. [Leading zeroes are not allowed.]

Solved by:   Nick McGrath, Andreas Abraham, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Claudio Meller, Naim Uygun, Ender Aktulga

* Cube Rearranger #2
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M3 to get N3, and where neither M nor M3 contains a 0.

Solved by:   Naim Uygun, Paul Cleary, P.M.A. Hakeem, Mark Rickert, Arthur Vause, Ender Aktulga

* Triple Rearranger
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, you can rearrange the digits of M2 to get N2, you can rearrange the digits of M3 to get N3, and where M does not contain the digit zero.

Solved by:   Ender Aktulga, Naim Uygun, Paul Cleary, Mark Rickert

* Rearranger #3
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M4 to get N4, and where neither M nor M4 contains a 0.

Solved by:   Paul Cleary, P.M.A. Hakeem, Naim Uygun, Mark Rickert, Arthur Vause, Ender Aktulga

* Rearranger #4
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M5 to get N5, and where neither M nor M5 contains a 0.

Solved by:   Paul Cleary, Naim Uygun, P.M.A. Hakeem, Mark Rickert, Arthur Vause, Ender Aktulga

** Rearranger #5
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M6 to get N6, and where neither M nor M6 contains a 0.

Solved by:   Arthur Vause, Paul Cleary, Naim Uygun, Ender Aktulga, Mark Rickert

** Rearranger #6
Find the smallest whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M7 to get N7, and where neither M nor M7 contains a 0.

Solved by:   Arthur Vause, Paul Cleary, Naim Uygun, Ender Aktulga, Mark Rickert

* Foreheads (contributed by Denis Borris )
A math wizard has a bag containing the digits 0 through 9, and has used six of them to stick two different three-digit perfect squares on the foreheads of Ann and Ben. Both Ann and Ben know this fact, but each person can see only the other person's number. The wizard asks Ann, "How many of the digits remaining in my bag can you exactly tell me?" Ann replies, "Two." Ben then says: "I know my number"
There are 3 possible 3-digit numbers on Ben's forehead. What are they?

Solved by:   Andreas Abraham, Mark Rickert, P.M.A. Hakeem, Claudio Meller, Naim Uygun, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga Send us an email to submit answers to these problems, to ask for help, or to submit new problems. We welcome your contributions. Be sure to change the \$ to an @ in our email address.