Mathematical puzzles involving the digits of a number
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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



* 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



* 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



* 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



* 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



### Powers of Digits (contributed by Paul Cleary)
       Let N be an integer with an even number of decimal digits, say abcd...yz. For how many N does N = ab cd ... yz? One example is 2592 = 25 92. Are there any others? [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



** 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



** 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



** 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 5k. [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



** 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



** 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 powers must be whole numbers.)
       How many such numbers exist?



** 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 all such families.

Solved by:   P.M.A. Hakeem



* 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



* 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



* 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



** 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



### 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?



** Consecutive (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



* 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



* 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



* 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



* 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 PK 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 PK.

Solved by:   Nick McGrath, Sudipta Das, P.M.A. Hakeem, Paul Cleary



* 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



** 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 10 are not allowed. For example, you could not add 8 to the digits of 81 to get 169.]

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 (1 solution)



** 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



* 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



*** 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 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)



** 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 (has 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



* 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



** 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 over an hour. 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



* 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



** 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



* 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



* 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. No computer is needed.]

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



* 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



** 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



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

Solved by:   Jens Kruse Andersen



** Treble Digits
       Find an 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



** 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



** 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



** 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



** 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



* 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



** 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



*** 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



*** 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



*** Peeling Primes #2 (contributed by Paul Cleary)
       What is the longest 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.




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