The Contest Center - Digit Puzzles

Mathematical puzzles involving the digits of a number

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DIGITS
The Contest Center
59 DeGarmo Hills Road
Wappingers Falls, NY 12590

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We will post the names of anyone who solves these digit 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.

* A Self-Describing Number
Find a 10-digit number that describes itself. The first digit will be the numbers of 1's in the number, the second digit will be the number of 2's, and so forth, through the ninth digit. The tenth digit will be the number of 0's.

Solved by: Nick McGrath, Ritwik Chaudhuri, Sudipta Das, Damian Carroll, Michael Burton, Marc Schegerin, Hashim Mooppan, Patrick J. Aber, Andreas Abraham, Audrey Droesch, Gaurav Agrawal, S. Preethi Sudharsha, Arijit Bhattacharyya, Jens Kruse Andersen, N.S. Kamra, Andrew C.J.O. Wandera, Jordan Canete, Patrick Vincent G. Aquino, Adhyas Avasthi, Abinash Pati, Sneh Sagar, Jean-Charles Desjardins, Swapnil Sonawane, Kyle Eberlin, Rahul Jagtap, Janaki Sivaramakrishnan, Susil Kumar Jena, P.M.A. Hakeem, Tan Lye Huat, Michael DeFranco, Pratheep Chellamuthu, Naim Uygun, Ionut-Zaharia Chirila, Selçuk Kumbasar, Turgay Yüktaş, Toprak Gök, Ender Aktulga, Kushal Khaitan, Abhay Menon, Ved Pant, Kaustuv Sengupta

* Even and Odd Products (contributed by Naim Uygun)
Find two different 5-digit numbers, each containing 5 different even digits, whose product is a square. Do the same for odd digits.

Solved by: Ender Aktulga, Paul Cleary, Turgay Yüktaş, Kaustuv Sengupta, P.M.A. Hakeem, Selçuk Kumbasar, Ionut-Zaharia Chirila

* Self-Contained (contributed by Naim Uygun)
Let N and P be positive integers, with 1<P≤50, such that N^P contains N. Find 1 example where N is a 9-digit pandigital number, and 3 examples where N is a 10-digit pandigital number.

Solved by: Ender Aktulga, Selçuk Kumbasar, Paul Cleary, Kaustuv Sengupta

* Embedded Digital Product (contributed by Naim Uygun)
Let N₀=348364879. The digital product of N, that is, the product of its nonzero digits, is P₀=3483648, which is contained within N₀. Find additional examples of numbers that contain their own digital products, with N₀<N₁<N₂<N₃<... and P₀<P₁<P₂<P₃<...

Solved by: Ender Aktulga (9 solutions), Paul Cleary (49 solutions), Turgay Yüktaş (154 solutions)

* 2×3 = 3×2 (contributed by Kaustuv Sengupta)
The 2×3 grid

2	6	4
2	0	0

has an interesting property: If you read horizontally, you get 264 and 200, where 264×200 = 52800. If you read vertically, you get 22, 60 and 40, where 22×60×40 = 52800. So the horizontal product is the same as the vertical product. Find another 2×3 grid with the same property using all 6 digits from 2 to 7.

Solved by: Naim Uygun, Ender Aktulga, P.M.A. Hakeem, Paul Cleary, Turgay Yüktaş, Selçuk Kumbasar, Ionut-Zaharia Chirila

* 2×4 = 4×2 (contributed by Naim Uygun)
The 2×3 grid

2	6	4
2	0	0

has an interesting property: If you read horizontally, you get 264 and 200, where 264×200 = 52800. If you read vertically, you get 22, 60 and 40, where 22×60×40 = 52800. So the horizontal product is the same as the vertical product. Find a 2×4 grid with the same property using 8 different digits.

Solved by: Ender Aktulga, Paul Cleary, Turgay Yüktaş, Selçuk Kumbasar, Kaustuv Sengupta, Ionut-Zaharia Chirila

*** Triangular Doublestring (contributed by Naim Uygun)
When the triangular number 6=3*4/2 is concatenated to itself, the result is a doublestring triangular number 66=11*12/2. Find the next example.

Solved by: Ahmet Saracoglu

** Integer Diagonal (contributed by Naim Uygun)
Find a rectangular prism whose edges and main diagonal have integer lengths, such that the squares of the edges are 10-digit pandigital numbers.

Solved by: Ender Aktulga, Paul Cleary, Turgay Yüktaş, P.M.A. Hakeem, Selçuk Kumbasar, Kaustuv Sengupta

* Conjoined Twins (contributed by Naim Uygun)
Find the smallest pair of twin primes that, concatenated, contain all 10 decimal digits.

Solved by: Paul Cleary, Ender Aktulga, Turgay Yüktaş, Selçuk Kumbasar, Kaustuv Sengupta, Ionut-Zaharia Chirila

** Five Pandigital Primes
Find five 11-digit pandigital primes P<Q<R<S<T that can be written in a column so that each of the first 10 columns of digits has 5 distinct digits. Find the set with smallest T.

Solved by: Paul Cleary, Ender Aktulga, Naim Uygun, Kaustuv Sengupta, Selçuk Kumbasar

** Eight Pandigital Primes
Find eight 11-digit pandigital primes P<Q<R<S<T<U<V<W that can be written in a column so that each of the first 10 columns of digits has 8 distinct digits. Find the set with smallest W.

Solved by: Ender Aktulga, Naim Uygun, Kaustuv Sengupta

*** Nine Pandigital Primes
Find nine 11-digit pandigital primes P<Q<R<S<T<U<V<W<X that can be written in a column so that each of the first 10 columns of digits has 9 distinct digits. Find the set with smallest X.

Solved by: Ender Aktulga, Naim Uygun, Ahmet Saracoglu, Kaustuv Sengupta

### Consecutive-Digit Primes
Find a 5×11 array of decimal digits such that each row, reading left to right is an 11-digit pandigital prime, and each of the first 10 columns contains 5 consecutive digits in some order.

* 123456 Triangle (contributed by Naim Uygun)
Find all integer-sided triangles (a,b,c), where each side is a permutation of N=123456 and the area is an integer. List your triangles with a<b<c.

Solved by: Ender Aktulga, Arthur Vause, Selçuk Kumbasar, Paul Cleary

* 12345678 Triangle (contributed by Arthur Vause)
Find all integer-sided triangles (a,b,c), where each side is a permutation of N=12345678 and the area is an integer containing only the digits 1,2,3,4,5,6,7,8. List your triangles with a<b<c.

Solved by: Naim Uygun, Ender Aktulga

* Pandigital Triangle
Find the smallest (in area) triangle where each side contains all of the digits from 0 to 9, and the area is an integer containing all of the digits 0 to 9 at least twice each.

** Pandigital Squared (contributed by Naim Uygun)
Three 10-digit pandigital numbers A, B and C satisfy A×B = C². Find the smallest and largest possible values for C.

Solved by: Paul Cleary, Ender Aktulga, Kaustuv Sengupta, Arthur Vause, Selçuk Kumbasar

** Pandigital Pythagorean Triangle
Three 10-digit pandigital numbers A, B and C satisfy A² + B² = C². Find the smallest and largest possible values for C.

Solved by: Naim Uygun, Paul Cleary, Ender Aktulga, Selçuk Kumbasar

* Reversible P and P² (based on an idea from Naim Uygun)
The prime P=13 reverses to the prime 31, and its square 169 reverses to 961, which is the square of 31. Find additional examples where R(P)² = R(P²). [Do not include trivial examples, like 101 or 10201. Do not take a solution, like 113, and simply stuff in zeroes, like 1103 and 11003.] Extra credit for any solution containing a digit larger than 3.

Solved by: Ender Aktulga (37 solutions), Paul Cleary (76 solutions + 7 extra credit), Turgay Yüktaş (18 solutions), Kaustuv Sengupta (13 solutions), Ionut-Zaharia Chirila (11 solutions)

** Powers of the Digits (with Naim Uygun)
Let N be an integer expressed in decimal. Let D(N) be the sum of the digits of N expressed in decimal. Let S(N) be the sum of the squares of the digits of N expressed in decimal. Let C(N) be the sum of the cubes of the digits of N expressed in decimal.

Find the smallest pandigital number N (that is, containing all of the digits 0 to 9) which contains D(N), S(N) and C(N).

Solved by: Ender Aktulga, Selçuk Kumbasar, Paul Cleary, Mark Rickert, Kaustuv Sengupta

* Descending Square (with Naim Uygun)
What is the largest square (not ending in 0) whose digits are in descending order, for example 441 and 961?

Solved by: Ender Aktulga, Selçuk Kumbasar, Turgay Yüktas, Paul Cleary, P.M.A. Hakeem, Kaustuv Sengupta

### Double Embedding
Find the smallest positive integer, N, such that N³ contains two copies of N, expressed in decimal.

* N² and N⁴
Find as many positive integers, N, not ending in 0, as you can, such that N² contains N and N⁴ contains N².

Solved by: Naim Uygun (4 solutions), Ender Aktulga (4 solutions), Paul Cleary (4 solutions), Selçuk Kumbasar (4 solutions), Kaustuv Sengupta (4 solutions)

* N, N² and N³
Find as many positive integers, N, not ending in 0, as you can, such that N³ contains both N and N².

Solved by: Naim Uygun (3 solutions), Ender Aktulga (3 solutions), Paul Cleary (3 solutions), Selçuk Kumbasar (3 solutions), Kaustuv Sengupta (3 solutions)

* N and N⁵ (contributed by Paul Cleary)
Find an infinite family of positive integers, N, not ending in 0, such that N⁵ contains two copies of N.

Solved by: Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Kaustuv Sengupta

* Cut Off the Tail (contributed by Naim Uygun)
Find at least two triangular numbers (numbers of the form N(N+1)/2) which include all 10 decimal digits, and which remain triangular when you delete the units digit.

Solved by: Ender Aktulga, Selçuk Kumbasar

### N and N⁵ (contributed by Paul Cleary)
Does there exist an integer, N, such that N⁵ contains three copies of N?

* Sum of 2 Squares (contributed by Naim Uygun)
Let A and B be two different whole squares, A < B, such that the set of decimal digits of A is the same as the set of decimal digits of B. Find the smallest and largest value of A+B which consists of 10 distinct digits.

Solved by: Ender Aktulga, Paul Cleary, Selçuk Kumbasar, Kaustuv Sengupta

* Same Digits (contributed by Naim Uygun)
If N is a decimal integer, let D(N) denote the set of its decimal digits. For example, D(3131663) is {1,3,6}. Find the first 3 positive integers such that D(N)=D(N²)=D(N³), other than the trivial solutions 1, 10, 100, etc.

Solved by: Paul Cleary, P.M.A. Hakeem, Selçuk Kumbasar, Turgay Yüktas, Toprak Göke, Ender Aktulga, Kaustuv Sengupta

** Two Pandigitals (contributed by Naim Uygun)
Find the smallest whole number N such that N² is the product of two distinct 10-digit pandigital numbers A and B.

Solved by: Ender Aktulga, Selçuk Kumbasar, Paul Cleary, Mark Rickert

** Three Pandigital Cube (contributed by Naim Uygun)
Find the smallest whole number N such that N³ is the product of three distinct 10-digit pandigital numbers A, B and C.

Solved by: Ender Aktulga, Selçuk Kumbasar, Paul Cleary, Kaustuv Sengupta

### 6 Squares (contributed by Naim Uygun)
The numbers 1 and 44 form squares when they are concatenated in either order, namely 144=12² and 441=21². Are there 3 integers A, B and C such that all 6 of their possible concatenations form squares? If not, what is the largest number of squares that can be formed by concatenating 3 integers?

** Eighth Powers (contributed by Naim Uygun)
Find two consecutive integers, A and B, with B=A+1, such that the decimal digits of A⁸ are a permutation of the decimal digits of B⁸. One example is A=5474857, B=5474858. Find additional examples.

Solved by: Ender Aktulga, P.M.A. Hakeem, Paul Cleary (11 solutions), Arthur Vause (112 solutions), Selçuk Kumbasar (316 solutions), Mark Rickert (119 solutions), Kaustuv Sengupta, Ionut-Zaharia Chirila

** Same Digits #2 (contributed by Paul Cleary)
Find the smallest integers M and N, where the decimal digits of M are the reverse of the digits of N, with M≠N, such that the digits of Mⁿ are a permutation of the digits of Nⁿ for n=2,3,4,...,24. For example, the digits of 1000201ⁿ are a permutation of the digits of 1020001ⁿ for n=2, 3 and 4.

Solved by: Arthur Vause, Selçuk Kumbasar, Ender Aktulga, Naim Uygun

* Square of a Palindrome (contributed by Naim Uygun)
There are integers which are both the concatenation of two consecutive integers and the square of a palindromic integer. Some examples are 528529 = 727², 4049540496 = 63636² and 20661152066116 = 4545454². Find additional examples.

Solved by: Ender Aktulga (3 solutions), Selçuk Kumbasar (7 infinite families), Turgay Yüktas (2 solutions), Paul Cleary (7 infinite families), P.M.A. Hakeem (2 solutions), Kaustuv Sengupta (3 solutions), Ionut-Zaharia Chirila (1 solution)

* Palindromic Squares (based on an idea from Naim Uygun who found 3 solutions)
The squares of 11, 22, 101, 202, ... are all palindromes with an odd number of digits. Find at least 2 integers whose squares are palindromes with an even number of digits.

Solved by: Ender Aktulga (3 solutions), Paul Cleary (3 solutions), P.M.A. Hakeem (2 solutions), Turgay Yüktas (3 solutions), Selçuk Kumbasar (4 solutions), Arthur Vause (3 solutions), Mark Rickert (5 solutions), Kaustuv Sengupta (4 solutions), Ionut-Zaharia Chirila (2 solutions)

* Two Palindromics (contributed by Naim Uygun)
What is the smallest whole number N such that N+123456 and N+12345678 are both palindromic? For example, the smallest whole number such that N+1234 and N+123456 are both palindromic is N=975445.

Solved by: Paul Cleary, Selçuk Kumbasar, Turgay Yüktas, Toprak Göke, Ender Aktulga, P.M.A. Hakeem, Ionut-Zaharia Chirila, Kaustuv Sengupta

* Double the Digits (based on an idea from Naim Uygun)
The binary integer 110101 has 2 zeroes and 4 ones. Its square 101011111001 has double that, namely 4 zeroes and 8 ones. Similarly, the ternary integer 20211 has 1 zero, 2 ones and 2 twos. Its square 1201102221 has double that, namely 2 zeroes, 4 ones and 4 twos. Find the first 3 decimal integers with this property.

Solved by: Paul Cleary, P.M.A. Hakeem, Selçuk Kumbasar, Turgay Yüktas, Toprak Göke, Ender Aktulga, Kaustuv Sengupta, Ionut-Zaharia Chirila

* Pell Equation (contributed by Paul Cleary)
Find positive integers X, Y and N satifying the Pell equation X² - N*Y² = 1 such that X, Y and N together contain all of the decimal digits 0 to 9 exactly once each. (The solution is unique when X,Y,N > 1.)

Solved by: P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Arthur Vause, Selçuk Kumbasar, Turgay Yüktas, Toprak Göke, Ahmet Saracoglu, Kaustuv Sengupta, Ionut-Zaharia Chirila

** Pell Equation #2 (contributed by Paul Cleary)
Find positive integers X, Y and N satifying the Pell equation X² - N*Y² = 1 such that X, Y and N together contain all of the decimal digits 0 to 9 exactly twice each.

Solved by: P.M.A. Hakeem (19 solutions), Arthur Vause (151 solutions), Selçuk Kumbasar (151 solutions), Ahmet Saracoglu (151 solutions), Ender Aktulga (98 solutions)

** 2⁸⁹ and 2⁹⁸ (contributed by Paul Cleary)
Find the smallest 89-digit integer which is a multiple of 2⁸⁹, and the smallest 98-digit integer which is a multiple of 2⁹⁸, both consisting solely of the digits 8 and 9.

Solved by: Arthur Vause, Ahmet Saracoglu, P.M.A. Hakeem, Selçuk Kumbasar, Ender Aktulga, Mark Rickert, Kaustuv Sengupta

*** Three Pandigitals
Find the smallest whole number N=ABC such that A, B and C each contain all of the digits 0 to 9 at least once, and N contains all of the digits 0 to 9 at least three times.

Solved by: Ahmet Saracoglu, Arthur Vause, Selçuk Kumbasar, Kaustuv Sengupta

* Digits of N³ (contributed by Paul Cleary)
Find a whole number N such that the digits of N combined with the digits of N² are exactly the digits of N³. (Extra credit: Find additional solutions, other than multiples of 10.)

Solved by: Naim Uygun (245 solutions), P.M.A. Hakeem (57 solutions), Arthur Vause (1000 solutions), Selçuk Kumbasar (1863 solutions), Ender Aktulga (1685 solutions), Turgay Yüktas (60 solutions), Mark Rickert (11956 solutions), Kaustuv Sengupta (60 solutions)

* Digits of N⁶ (contributed by Naim Uygun)
Find a whole number N such that the digits of N combined with the digits of N² and the digits of N³ are exactly the digits of N⁶. (Extra credit: Find additional solutions, other than multiples of 10.)

Solved by: Paul Cleary (256 solutions), Naim Uygun (12 solutions), P.M.A. Hakeem (8 solutions), Ender Aktulga, Turgay Yüktas (2 solutions), Selçuk Kumbasar (1682 solutions), Mark Rickert (256 solutions), Kaustuv Sengupta (3 solutions)

** Prime Cube (contributed by Naim Uygun)
Find the smallest prime number whose cube contains each of the digits 0 to 9 at least three times.

Solved by: Ender Aktulga, Paul Cleary

* Once, Twice, Thrice (contributed by Naim Uygun)
Find the smallest 11-digit positive integer N that contains each of the digits 0 to 9 at least 1 time, whose square contains each of the digits 0 to 9 at least two times, and whose cube contains each of the digits 0 to 9 at least three times.

Solved by: Ender Aktulga, Paul Cleary

* 20 and 40
Find the smallest 20-digit integer N which contains each of the digits 0 to 9 twice, whose square is a 40-digit number which contains each of the digits 0 to 9 four times.

Solved by: Paul Cleary, Ender Aktulga, Selçuk Kumbasar, Naim Uygun

* 20 and 60
Find the smallest 20-digit integer N which contains each of the digits 0 to 9 twice, whose cube is a 60-digit number which contains each of the digits 0 to 9 six times.

Solved by: Paul Cleary, Ender Aktulga, Selçuk Kumbasar, Naim Uygun

* 20 and 80
Find the smallest 20-digit integer N which contains each of the digits 0 to 9 twice, whose fourth power is an 80-digit number which contains each of the digits 0 to 9 eight times.

Solved by: Paul Cleary, Ender Aktulga, Selçuk Kumbasar, Naim Uygun

** 30 and 60
Find the smallest 30-digit integer N which contains each of the digits 0 to 9 three times, whose square is a 60-digit number which contains each of the digits 0 to 9 six times.

Solved by: Ahmet Saracoglu, Naim Uygun, Selçuk Kumbasar, Ender Aktulga

** 30 and 90 (contributed by Naim Uygun)
Find the smallest 30-digit integer N which contains each of the digits 0 to 9 three times, and whose cube is a 90-digit number which contains each of the digits 0 to 9 nine times.

Solved by: Ahmet Saracoglu, Ender Aktulga

*** 30, 60 and 90 (contributed by Naim Uygun)
Find the smallest 30-digit integer N which contains each of the digits 0 to 9 three times, whose square is a 60-digit number which contains each of the digits 0 to 9 six times, and whose cube is a 90-digit number which contains each of the digits 0 to 9 nine times.

* Big Palindrome (based on an idea from Naim Uygun)
Find the smallest integer AB=C where A and B are both 10-digit pandigital numbers, that is, contain all of the 10 digits 0 to 9 once each, and the product C is a 20-digit palindrome.

Solved by: Paul Cleary, Ahmet Saracoglu, Ahsen Canat, P.M.A. Hakeem, Selçuk Kumbasar

* Last Digits of N² (contributed by Paul Cleary)
Prove that there are infinitely many integers N such that N² ends with exactly the decimal digits of N. For example, 25² is 625, which ends in 25.

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

### Powers of Digits (contributed by Paul Cleary)
Let N be a positive integer with an even number of decimal digits, say abcd...yz. For how many N does N = a^b c^d ... y^z? One example is 2592 = 2⁵ 9². [If 0⁰ occurs, take its value as 1.]

** Powers of Digits #2 (contributed by Amogh Keni)
Let N be a positive integer with an even number of decimal digits, say abcd...yz. For how many N does a^b + c^d + ... + y^z? equal zy...dcba, that is, N with its digits reversed? One example is N=52, since 5² = 25. [If 0⁰ occurs, take its value as 1. Do not include any solutions ending with 00.]

Solved by: P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Kaustuv Sengupta

### Powers of Digits #3 (based on an idea from Amogh Keni)
Let M and N be distinct positive integers with an even number of decimal digits, say M=abcd...ef and N=ghij...kl. Find pairs of M and N such that N = a^b+c^d+...+e^f and M = g^h+i^j+...+k^l. [If 0⁰ occurs, take its value as 1.]

* All Combos
What is the smallest integer N such that N! expressed in decimal contains (A) all of the digits 0 to 9, (B) all of the 2-digit combinations 00, 01, ..., 99, (C) all of the 3-digit combinations 000, 001, ..., 999.

Solved by: Paul Cleary, P.M.A. Hakeem, Richard Zapor, Naim Uygun, Arthur Vause, Selçuk Kumbasar, Ender Aktulga, Turgay Yüktas, Mark Rickert, Kaustuv Sengupta

** Binary Decimal (contributed by Paul Cleary)
Find the smallest integer N such that the decimal representation of N contains only the digits 0 and 1, and there are 8 distinct integers A,B,C,D,E,F,G,H where the 8 quotients N/A, N/B, N/C, ..., N/H are all pandigital integers. That is, each quotient contains each of the digits 0 to 9 exactly once. For example, 1001100111111 / 373 = 2683914507.

Solved by: Nick McGrath, P.M.A. Hakeem, Naim Uygun, Arthur Vause, Selçuk Kumbasar, Ender Aktulga, Kaustuv Sengupta

** Totally Even (contributed by Steve Spindler)
Call a positive integer totally even if all of its decimal digits are even. Show that for every positive integer N there is a multiple of N which is totally even.

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

** Powers of 5
Call a positive integer totally odd if all of its decimal digits are odd. Show that for any positive integer k there is a k-digit totally odd integer which is a multiple of 5^k. [This problem is from the 2003 USA Mathematical Olympiad. It is included here because the result may be useful for the following problem.]

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

** Totally Odd (contributed by Steve Spindler)
Call a positive integer totally odd if all of its decimal digits are odd. Show that for every odd integer N there is a multiple of N which is totally odd.

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

** Digit Powers
There are many numbers that can be expressed as the sum of powers of their digits. For example 2³+4²=24, 4²+3³=43, 6²+3³=63, 8¹+9²=89, 1¹+3²+5³=135, and so forth. (The exponents must be positive integers.)
How many such numbers exist?

Solved by: P.M.A. Hakeem

** Rising Numbers
We will call an integer of 2 or more digits "rising" if each of its digits, starting at the second digit, is at least as large as the digit to its left. For example 123 and 233379999 are rising numbers. There are many pairs of rising numbers whose product is also a rising number, for example 12×13=156. There several infinite families of such pairs, both having the same number of digits, for example 3×4=12, 33×34=1122, 333×334=111222, etc. Find at least 20 such families.

Solved by: Frank Rubin (10 doubly-infinite families), P.M.A. Hakeem (63 families), Amogh Keni (a triply-infinite family), Naim Uygun (65 families), Ender Aktulga (73 families)

* Digital Sum Triples
The digital sum S(N) of an integer N is the sum of its decimal digits. So S(128)=11. Three positive integers A, B and C such that A = S(B)S(C), B = S(A)S(C), and C = S(A)S(B) are called a digital sum triple, for example A=B=C=81. Find all such triples.

Solved by: Andreas Abraham, Steve Spindler, Sudipta Das, Shyam Sunder Gupta, P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Kaustuv Sengupta, Ionut-Zaharia Chirila

* Digital Product Triples #1
The digital product P(N) of an integer N is the product of its decimal digits. For example, P(128)=16. Find all sets of three positive integers A, B and C such that A = P(B)+P(C), B = P(A)+P(C), and C = P(A)+P(B).

Solved by: Jean Jacquelin, Andreas Abraham, Sudipta Das, Shyam Sunder Gupta, P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Turgay Yüktas, Selçuk Kumbasar, Kaustuv Sengupta, Ionut-Zaharia Chirila

* Digital Product Triples #2
The digital product P(N) of an integer N is the product of its decimal digits. So P(128)=16. Find all sets of distinct positive integers A and B such that A = P(B)P(C) and B = P(A)P(C) for some integer C. For each pair, A and B, give the lowest possible value for C.

Solved by: Nick McGrath, Jean Jacquelin, P.M.A. Hakeem, Ender Aktulga, Naim Uygun

** Digital Product Triples #3
The digital product P(N) of an integer N is the product of its non-zero decimal digits. So P(128)=16. A Digital Product Triple is a set of positive integers A, B and C such that A = P(B)P(C), B = P(A)P(C) and C = P(A)P(B). One example is A=25, B=C=50. Find a digital product triple where A, B and C are all different.

Solved by: Lee Morgenstern, P.M.A. Hakeem

* Digital Sums (contributed by Nick McGrath)
The digital sum of an integer is the sum of its digits. Find three positive integers such that the digital sum of the sum of any two of them is less than 5, but the digital sum of the sum all three is greater than 50.

Solved by: Jean Jacquelin, Andreas Abraham, Jens Kruse Andersen, P.M.A. Hakeem, Mark Rickert

### Two Concatenations
The squares 4 and 9 can be concatenated to form 49, which is another square. Remarkably, the square of 4 and the square of 9 can be concatenated to form 1681, which is also square. Are there other examples (besides 4900, 490000, etc.)? Are there any examples using 3 squares?

* Three Concatenations (Naim Uygun and Paul Cleary)
Find three whole numbers A, B and C such that the concatenations AB, AC and BC are all squares, and the sum A+B+C is also a square.

Solved by: P.M.A. Hakeem, Ender Aktulga, Selçuk Kumbasar, Kaustuv Sengupta

** Consecutive Pandigitals (contributed by Denis Borris)
Find a sequence of consecutive integers, a, a+1, a+2, ..., b such that b is a square and that [a+(a+1)+(a+2)+...+(b-1)]b and [(a-1)+a+(a+1)+(a+2)+...+(b-1)]b are both 10-digit numbers containing all of the digits 0 to 9.

Solved by: Andreas Abraham, Sudipta Das, Mark Rickert, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Ahmet Saracoglu, Ender Aktulga, Selçuk Kumbasar, Kaustuv Sengupta

* Factorial Factored (contributed by Sudipta Das)
Write the factorization of N! in terms of primes and powers. For example, 6! = 2⁴ 3² 5¹. (A) What is the smallest N for which the prime factors of N! and the exponents together contain all of the digits 0 to 9? (B) What is the smallest N for which the prime factors of N! contain all of the digits 0 to 9, and the exponents also contain all of the digits 0 to 9?

Solved by: Nick McGrath, Andreas Abraham, Carlos Rivera, Jens Kruse Andersen, P.M.A. Hakeem, Claudio Meller, Paul Cleary, Richard Zapor, Ender Aktulga, Turgay Yüktas, Selçuk Kumbasar, Mark Rickert, Naim Uygun

* Factors and Exponents (contributed by Carlos Rivera)
Write the factorization of N in terms of primes and powers. For example, 600 = 2³ 3¹ 5². (A) What is the smallest N for which the prime factors of N and the exponents together contain all of the digits 0 to 9? (B) What is the smallest N for which the prime factors of N contain all of the digits 0 to 9, and the exponents also contain all of the digits 0 to 9?

Solved by: Jean Jacquelin, Jens Kruse Andersen, P.M.A. Hakeem, Paul Cleary, Ender Aktulga

* Divisible by Squares (contributed by Sudipta Das)
Find the largest integer A such that the first K digits of A form an integer divisible by K² for K=1,2,3,...,N where N is the number of digits in A.

Solved by: Nick McGrath, Carlos Rivera, Gaurav Agrawal, Andreas Abraham, Mark Rickert, Janaki Sivaramakrishnan, P.M.A. Hakeem, Paul Cleary, Naim Uygun, Ender Aktulga, Selçuk Kumbasar

* Divisible by Primes (contributed by Carlos Rivera)
Find the largest integer A such that the first K digits of A form an integer divisible by P_K the K-th prime for K=1,2,3,...,N where N is the number of digits in A. Similarly, find the largest integer where the last K digits are divisible by P_K.

Solved by: Nick McGrath, Sudipta Das, P.M.A. Hakeem, Paul Cleary, Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Mark Rickert

* Divisible by Length (contributed by Paul Cleary)
Find the largest integer A such that for K=1,2,3,...,length(A) the first K digits of A form an integer divisible by K. For example, for 56165, 5 is divisible by 1, 56 is divisible by 2, 561 is divisible by 3, 5616 is divisible by 4 and 56165 is divisible by 5.

Solved by: P.M.A. Hakeem, Selçuk Kumbasar, Ender Aktulga, Mark Rickert, Kaustuv Sengupta

** Digit Increments (contributed by Sudipta Das)
The following squares remain squares when you add 1 to each of their digits: 0, 25, 2025, 13225, 4862025, 60415182025, and 207612366025. (For example, 25 becomes 36.) Find additional cases. [Digit sums greater than 9 are not allowed, so the smaller square cannot contain the digit 9.]

Solved by: Nick McGrath (5 solutions), Carlos Rivera (5 solutions), Andreas Abraham (27 solutions), Jens Kruse Andersen (1 solution), Mark Rickert (2 solutions), Paul Cleary (70 solutions), P.M.A. Hakeem (13 solutions), Claudio Meller (6 solutions), Naim Uygun (5 solutions), Arthur Vause (70 solutions), Ionut-Zaharia Chirila (found an infinite family), Ender Aktulga (2 solutions), Kaustuv Sengupta (6 solutions)

** Digit Increments #2 (contributed by Paul Cleary)
For which digits d is it possible to add d to every digit of a square and get another square? For example, adding 3 to each digit of 16 gives 49. For which digits d are there infinitely many such squares? [Digit sums greater than 9 are not allowed. For example, you could not add 8 to the digits of 81 to get 169.]

Solved by: P.M.A. Hakeem, Naim Uygun, Arthur Vause (2 infinite families), Ender Aktulga

* P/S/C
Find the smallest whole number whose decimal digits can be rearranged to form a prime, a square, and a cube.

Solved by: Marc Schegerin, Hashim Mooppan, Sudipta Das, Nick McGrath, Andreas Abraham, Denis Borris, Mark Rickert, Susil Kumar Jena, P.M.A. Hakeem, Richard Zapor, Naim Uygun, Paul Cleary, Ionut-Zaharia Chirila, Selçuk Kumbasar, Kushal Khaitan, Ender Aktulga, Abhay Menon

*** P/S/C #2
Show that there are infinitely many whole numbers whose decimal digits can be rearranged to form a prime, a square, and a cube in a unique way. For example, the unique way to arrange 222257 is prime=2, square=225 and cube=27.

** End to Middle #1 (contributed by Sudipta Das)
Find the first 3 (or more) squares, such that when you move the last digit to the middle it becomes a larger square. For example 169 becomes 196.

Solved by: Nick McGrath (5), Jens Kruse Andersen (11), Paul Cleary (5), P.M.A. Hakeem (4), Claudio Meller (3), Naim Uygun (5), Ender Aktulga (3)

** End to Middle #2 (contributed by Sudipta Das)
Find the first 3 squares, such that when you move the last digit to the middle it becomes a smaller square. For example 196 becomes 169.

Solved by: Jean Jacquelin (found an infinite family of solutions, and has written a program capable of generating solutions up to 1000 digits), Jens Kruse Andersen (found the same family, plus 6 additional solutions), Claudio Meller, Ahmet Saracoglu, P.M.A. Hakeem (found an infinite family)

* Square Prime (contributed by Nick McGrath)
Find the smallest square containing all of the digits from 0 to 9 such that when you reverse the digits the number is prime. For example, the square 361 reverses to 163 which is prime.

Solved by: Carlos Rivera, Andreas Abraham, Shyam Sunder Gupta, Mark Rickert, P.M.A. Hakeem, Paul Cleary, Claudio Meller, Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Ionut-Zaharia Chirila

** The Maths Wizard (contributed by Sudipta Das)
       Tom, Dick and Harry are the prisoners of the Maths Wizard. The Wizard shows them a box with each of the digits 1 through 9 exactly TWICE, and tells them he will place 6 of these 18 digits on each one's forehead to form a perfect square. Each person can see the numbers on the others' foreheads, but not on his own.
       The Wizard announced, "I will set each of you free as soon as you can tell me the number on your forehead. Nobody is allowed to communicate with any other person in any way. You get only one guess each."
       Tom, Dick and Harry sat in silence for several hours, filling up pages of calculations. Getting bored, the Wizard said, "Hey, I had thought you were clever guys. All right, I'll give each one of you just one hint. Tom, the square root of your number is a palindrome. Dick, the square root of your number is not a palindrome. Harry, any two adjacent digits of your number differ at most by 3."
       And then the Wizard's castle echoed, "GOT IT !!!" as they all shouted together. What were the numbers on each person's forehead?

Solved by: Nick McGrath, Denis Borris, P.M.A. Hakeem, Naim Uygun, Kushal Khaitan, Ender Aktulga, Kaustuv Sengupta

* Sum of Powers (contributed by Ritwik Chaudhuri)
Find the last decimal digit of the sum 1¹ + 2² + 3³ + ... + 2001²⁰⁰¹

Solved by: Sudipta Das, Nick McGrath, Carlos Rivera, Rakesh Kumar Banka, Amitayu Pal, Marc Schegerin, Hashim Mooppan, Andreas Abraham, Arijit Bhattacharyya, Jens Kruse Andersen, Mark Rickert, Janaki Sivaramakrishnan, Susil Kumar Jena, P.M.A. Hakeem, Tan Lye Huat, Claudio Meller, Naim Uygun, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga, Amogh Keni, Paul Cleary, Kaustuv Sengupta, Turgay Yüktaş

** Mileage
My car has a 5-digit odometer, which measures the miles since the car was built, and a 3-digit trip meter, which measures the miles since I last set it. Every so often, one or both of the readings is a palindrome, that is, it reads the same forwards and backwards, like 262 or 37173. The meters reset to 000 after 999 and to 00000 after 99999.
The current readings are 123 and 12345. Assuming that I do not reset the trip meter, when will both readings next be palindromes? What reading could they have so that it will take the most miles until they are both palindromes again? Is there any reading they could have so that they will NEVER both become palindromes?

Solved by: Aydin Gurel, Sudipta Das, Marc Schegerin, P.M.A. Hakeem, Ender Aktulga

* Remove Zeros (contributed by Sudipta Das)
Let N be a positive integer, and let M be the smaller integer formed by removing all of the 0 digits from N. For example, if N is 1030 then M would be 13. (A) Find the smallest N for which M evenly divides N, and N/M does not end in 0. (B) Find the smallest N for which M evenly divides N, and N/M contains the same digits as M. (C) Find the smallest N for which M evenly divides N, and both M and N/M are prime.

Solved by: Nick McGrath, Andreas Abraham, P.M.A. Hakeem, Paul Cleary, Naim Uygun, Ender Aktulga, Selçuk Kumbasar

* Factorials
The factorial of N, which is denoted N!, is the product of the integers 1×2×3×...×N. For example 4! is 1×2×3×4, or 24. What decimal digit occurs most often in the sequence of factorials from 77777! to 77877! inclusive? [This can be solved by logic. You do not need to compute any factorials.]

Solved by: Colin Bown, Carlos Rivera, Ritwik Chaudhuri, William Alber, Hashim Mooppan, Marc Schegerin, Sudipta Das, Andreas Abraham, Jens Kruse Andersen, Shyam Sunder Gupta, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Kushal Khaitan, Ender Aktulga, Selçuk Kumbasar, Kaustuv Sengupta

* Prime Factorials (contributed by Sudipta Das)
The digits of 2! have only one arrangement, namely 2, which is a prime. Can you find the next integer N such that the digits of N! can be rearranged to form a prime?

Solved by: Ritwik Chaudhuri, Nick McGrath, Carlos Rivera, Marc Schegerin, Hashim Mooppan, Andreas Abraham, Jens Kruse Andersen, Shyam Sunder Gupta, Mark Rickert, Arijit Bhattacharyya, P.M.A. Hakeem, Claudio Meller, Paul Cleary, Naim Uygun, Ionut-Zaharia Chirila, Kushal Khaitan, Ender Aktulga, Amogh Keni, Selçuk Kumbasar, Kaustuv Sengupta

** Auto-Power
       Define the auto-power sum of a positive integer to be the sum of its decimal digits, each digit taken to its own power. For example, the auto-power sum of 32 is 3³+2² = 27+4 = 31. Call a positive integer an auto-power integer if the number is equal to its auto-power sum. (For the purposes of this puzzle, assume 0⁰ to be 0.) Then 0 and 1 are trivial examples of auto-power integers. The number 3435 is also an auto-power number because 3³+4⁴+3³+5⁵ = 27+256+27+3125 = 3435.
       Show that there are only a finite number of auto-power integers.
       Now generalize the definition of auto-power sum to be any of the sums formed by taking one or more digits at a time, raising each of these numbers to its own power, and adding them. For example the auto-power sums of 345 would be 3³+4⁴+5⁵, 34³⁴+5⁵, 3³+45⁴⁵, and 345³⁴⁵. Call a positive integer an auto-power integer if it is equal to any of its auto-power sums. For example,

    20²⁰ + 9⁹ + 7⁷ + 1¹ + 5⁵ + 20²⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ + 0⁰ +
    4⁴ + 0⁰ + 5⁵ + 8⁸ + 5⁵ + 4⁴ + 7⁷ + 3³ + 3³
= 104857600000000000000000000 + 387420489 + 823543 + 1 + 3125 +
    104857600000000000000000000 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 +
    256 + 0 + 3125 + 17600759 + 3125 + 256 + 823543 + 27 + 27
= 20 97152 00000 00000 04058 54733

       Show that, using this more general definition, there are an infinite number of auto-power integers. [For full credit, you should find a rich set of such numbers, not merely the minimum set satisfying the conditions.]

Solved by: James Layland, Gilles Ravat, William Alber, Nick McGrath

* Descriptor Sequence
The descriptor sequence is a sequence of numbers in which the digits of each number describe the preceding number. The first number is 1. This number consists of one 1, so the second number is 11 (that is, one-one). This consists of two 1's, so the third term is 21. This consists of one 2 and one 1, so the fourth term is 1211. The first six numbers in the sequence are

1, 11, 21, 1211, 111221, 312211.

Show that no digit greater than 3 ever occurs, and that the string 333 never occurs.

Solved by: Matthew Ender, Hrishikesh Nene, William Alber, Marc Schegerin, Sudipta Das, Andreas Abraham, S. Preethi Sudharsha, Jens Kruse Andersen, Mark Rickert, Arijit Bhattacharyya, P.M.A. Hakeem, Ionut-Zaharia Chirila, Ender Aktulga, Kaustuv Sengupta

** Descriptor Sequence #2
Let the descriptor sequence be as defined above. Show that the string 13113 never occurs.

Solved by: Jens Kruse Andersen, Ionut-Zaharia Chirila

** Treble Digits
Find the smallest integer N such that N and N² together contain all 10 digits from 0 through 9 exactly 3 times each.

Solved by: Carlos Rivera, Sudipta Das, Nick McGrath, Andreas Abraham, Jens Kruse Andersen, Denis Borris, Mark Rickert, P.M.A. Hakeem, Paul Cleary, Naim Uygun, Ahmet Saracoglu, Ender Aktulga, Selçuk Kumbasar

** Treble Treble
Find the smallest whole number N such that N contains all of the digits from 0 through 9, N² contains all of the digits at least twice each, and N³ contains all of the digits at least three times each.

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

** Once and Twice (contributed by Naim Uygun)
Find the smallest whole number N such that N contains all of the digits from 0 through 9, and N² contains all of the digit pairs 00, 11, 22, ..., 99.

Solved by: Turgay Yüktas, Selçuk Kumbasar, Ender Aktulga

*** Pairs and Triples
Find the smallest whole number N such that N contains all of the digits from 0 through 9, N² contains all of the digit pairs 00, 11, 22, ..., 99, and N³ contains all of the digit triples 000, 111, 222, ..., 999.

Solved by: Adolph Mystein, Ahmet Saracoglu

** Reversible Square (submitted by Sudipta Das)
Find the smallest square containing all of the digits from 0 to 9, such that when you reverse the digits the result is also a square. (For example, the square 169 reversed is 961, which is also a square.)

Solved by: Carlos Rivera, Nick McGrath, Andreas Abraham, Jens Kruse Andersen, Shyam Sunder Gupta, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Arthur Vause, Ender Aktulga, Selçuk Kumbasar, Turgay Yüktas, Mark Rickert, Ionut-Zaharia Chirila

** Complementary Squares (submitted by Sudipta Das)
Two squares of the same length are complementary if the digits in corresponding positions in the 2 squares add to either 0 or 10. The first 4 such pairs without trailing zeroes are (1, 9), (21609, 89401), (22801, 88209), and (299209, 811801).
Find the next such pair, or prove that no other pair exists.

Solved by: Frank Rubin, Nick McGrath, Jens Kruse Andersen, Naim Uygun, Paul Cleary, Ender Aktulga, Kaustuv Sengupta

** Reversible Prime (submitted by Nick McGrath)
Find the smallest prime containing all of the digits from 0 to 9, such that when you reverse the digits the result is also a prime. (For example, the prime 179 reversed is 971, which is also a prime.)

Solved by: Carlos Rivera, Jens Kruse Andersen, Shyam Sunder Gupta, Mark Rickert, P.M.A. Hakeem, Claudio Meller, Paul Cleary, Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Ionut-Zaharia Chirila

** Permutable Prime (submitted by Sudipta Das)
Start with a prime. Delete one decimal digit and arrange the remaining digits to form a new prime. Then delete another digit, and so forth. What is the smallest prime you can start with if (A) you delete the digits 0,1,2,3,4,5,6,7,8,9 in any order? (B) you delete the digits 0,1,2,3,4,5,6,7,8,9 in that order?

Solved by: Nick McGrath, Mark Rickert, P.M.A. Hakeem, Paul Cleary, Ender Aktulga, Selçuk Kumbasar

* Pan-Permutable Prime
If you permute the digits of the prime 13 you get 31, which is also prime. If you permute the digits of the prime 337 you get 373 and 733, which are also prime. However, if you permute the digits of 379, some of the permutations are not prime, namely 793=13x61 and 973=7x139. What is the largest integer (containing at least 2 distinct digits) such that every permutation of its digits is prime?

Solved by: Nick McGrath, Paul Cleary, P.M.A. Hakeem, Naim Uygun, Ender Aktulga, Selçuk Kumbasar, Ionut-Zaharia Chirila

** Permutable Square (submitted by Sudipta Das)
Start with a square. Delete one decimal digit and arrange the remaining digits to form a new square. Then delete another digit, and so forth. What is the smallest square you can start with if (A) you delete the digits 0,1,2,3,4,5,6,7,8,9 in any order? (B) you delete the digits 0,1,2,3,4,5,6,7,8,9 in that order?

Solved by: Nick McGrath, P.M.A. Hakeem, Ender Aktulga, Selçuk Kumbasar

*** Growing Primes #1
Start with any prime. Append a decimal digit to make another prime. Append another decimal digit to make a third prime. And so forth. For example, if you start with 3, you can append 1 to make 31, then append another 1 to make 311, append 9 to make 3119, and append 3 to make 31193.
Show that no matter what prime you start with and what digits you append each such sequence always ends. That is, there comes a point where there is no digit that can be appended to form another prime. Conversely, show that for any N there is always a sequence of N such primes.

*** Growing Primes #2
Let A and B be any sequence of digits in base N, that is, they are integers expressed in base N, possibly with leading zeros. If gcd(A,B)=1 and gcd(B,N)=1 prove or disprove that the sequence AB, ABB, ABBB, ... must contain a prime. For example, if A=5 and B=1 in base 10, then 51=3x17, 511=7x73, 5111=19x269, 51111=3x3x3x3x631, but 511111 is prime.

Solved by: Carlos Rivera, Jens Kruse Andersen, Arthur Vause

** Peeling Primes (based on an idea from Carlos Rivera)
The prime 1366733339 can be peeled down to a single digit by removing one digit at a time from either end to make a sequence of primes 136673333, 36673333, 3667333, 667333, 66733, 6673, 673, 67, 7. What is the largest integer for which this is possible?

Solved by: Carlos Rivera, Sudipta Das, Jens Kruse Andersen, Paul Cleary, Ahmet Saracoglu, Arthur Vause, Selçuk Kumbasar, Mark Rickert, Ender Aktulga

*** Peeling Primes #2 (contributed by Paul Cleary)
What is the largest palindromic prime which remains prime as the two digits at each end are peeled off, one pair at a time, until only a 3-digit prime is left? For example, from the palindromic prime 30961016903, peel off the end digits 30 and 03 to get the palindromic prime 9610169, then peel off 96 and 69, to leave the prime 101.

Solved by: Arthur Vause, Ahmet Saracoglu, Selçuk Kumbasar, Mark Rickert, Ender Aktulga, Kaustuv Sengupta

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