The Contest Center - Mathematical Puzzles

Puzzles involving squares, cubes and higher powers

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SQUARES and POWERS
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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 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

* 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

### 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?

* 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

* 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

* 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

* 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

* 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

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

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.]

Power Digits
Let N=d₁d₂d₃...d_n be an n-digit decimal number, with n>1. Form the sum
S(N) = d₁ⁿ + d₂ⁿ + d₃ⁿ + ... + d_nⁿ
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)

** 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, such as 11000222244. We will list the number of squares found by each solver.

Solved by: Gerald Harrison(9), Sudipta Das(10), Andreas Abraham (5, plus an infinite family), Bipin Kumar (4), Mark Rickert (4)

** 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

** 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

*** 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

* 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

** 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

*** 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 12391239 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.

* 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.

** 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.

** 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

*** 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

** 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

* 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

* 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

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