**3.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**7695**

3 is the only prime 1 less than a perfect square.

**4.**Problem: , proposed by

**Ah**

**met Arduc**

4 is the smallest number of colors sufficient to color all planar maps.

**7.**Problem: , proposed by

**Ah**

**met Arduc**

7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass.

**9.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**29D9**

9 is the maximum number of cubes that are needed to sum to any positive integer.

**10.**Problem: , proposed by

**Ah**

**met Arduc**

10 is the only triangular number which is also the sum of 2 consecutive square odd numbers.

**11.**Problem: , proposed by

**Ah**

**met Arduc**

11 is the only palindromic prime with an even number of digits.

**12.**Problem: , proposed by

**Ah**

**met Arduc**

12 is the largest known even number expressible as the sum of two primes in one way.

**14.**Problem: , proposed by

**Ah**

**met Arduc**

14 is the smallest even number n with no solutions to φ(m) = n.

**19.**Problem: , proposed by

**Ah**

**met Arduc**

351 is the sum of five consecutive prime numbers: 61 + 67 + 71 + 73 + 79.

**20.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**8632**

1017 is the smallest number whose square contains 7 different digits.

**21.**Problem: , proposed by

**Ah**

**met Arduc**

21 is the smallest number of distinct squares needed to tile a square.

**24.**Problem: , proposed by

**Ah**

**met Arduc**

The product of 4 consecutive numbers n(n+1)(n+2)(n+3) is divisible by 24.

**25.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**7695**

25 is the smallest square number that can be written as a sum of 2 consecutive squares.

**27.**Problem: , proposed by

**Ah**

**met Arduc**

27 is the largest integer which is the sum of the digits of its cube.

**30.**Problem: , proposed by

**Ah**

**met Arduc**

30 is the largest number such that every smaller coprime to it is prime.

**31.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**A031878**.The sum of the first 31 odd primes is a perfect square.

**32.**Problem: , proposed by

**Ah**

**met Arduc**

32 the largest known power with all decimal digits being prime.

**33.**Problem: , proposed by

**Ah**

**met Arduc**

33 is the largest number that is not a sum of distinct triangular numbers.

**36.**Problem: , proposed by

**Ah**

**met Arduc**

36 is the smallest number which is the sum of pairs of distinct odd primes in four ways.

**37.**Problem: , proposed by

**Ah**

**met Arduc**

37 is the maximum number of 5th powers needed to sum to any number.

**38.**Problem: , proposed by

**Ah**

**met Arduc**

38 is the largest known even number which can be represented as sum of two distinct primes in only one way.

**39.**Problem: , proposed by

**Ah**

**met Arduc**

39 is the smallest number which has 3 different partitions into 3 parts with the same product.

**41.**Problem: , proposed by

**Ah**

**met Arduc**

41 is the lowest number whose cube is the sum of 3 cube numbers in 2 different ways.

**42.**Problem: , proposed by

**Ah**

**met Arduc**

42 is the he smallest perfect square that is the mean of two cubed twin primes.

**43.**Problem: , proposed by

**Ah**

**met Arduc**

43 is the smallest non-palindromic prime which on subtracting its reverse gives a perfect square.

**44.**Problem: , proposed by Amirul Faiz Abdul Muthalib

44 is the number of derangements of 5 objects.

**45.**Problem: , proposed by Amirul Faiz Abdul Muthalib

45 is both a Kaprekar and a triangular number.

**46.**Problem: , proposed by Amirul Faiz Abdul Muthalib, Tip: Key Fact(s):

**7695**

46 is the number of chromosomes all human beings have in every cell of their body.

**47.**Problem: , proposed by Amirul Faiz Abdul Muthalib

47 is the largest number of cubes that cannot tile a cube.

**factorial base of numeration**is defined as $a_1+a_2⋅2!+a_3⋅3!+...+a_n⋅n!$ where $a_1,a_2,...,a_n$ are integer coefficients such that $0≤a_k≤k$. Thus,

**1971**can be written in factorial base of numeration as $$1+1⋅2!+0⋅3!+2⋅4!+4⋅5!+2⋅6!$$ where the sum of the coefficients is

**10**. What is the sum of all coefficients used in factorial base of numeration to write all numbers from

**1**to

**2017**? Ah Math

**49.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**7695**

49 is the smallest number with exactly 8 representations as a sum of three distinct primes.

**50.**Problem: , proposed by Amirul Faiz Abdul Muthalib

50 is the smallest number that can be written as the sum of 2 squares in 2 distinct ways.

**52.**Problem: , proposed by Amirul Faiz Abdul Muthalib

52 is the 5th Bell number.

**54.**Problem: , proposed by Amirul Faiz Abdul Muthalib

54 is the smallest number that can be written as the sum of 3 squares in 3 ways.

**55.**Problem: , proposed by

**Ah**

**met Arduc**

55 is the largest triangular number in the Fibonacci sequence.

**59.**Problem: , proposed by Amirul Faiz Abdul Muthalib

Watch first S2E2 of NUMB3RS (the television crime drama): Better or Worse.

**62.**Problem: , proposed by

**Ah**

**met Arduc**

62 is the smallest number that can be written as the sum of 3 distinct squares in 2 ways.

**64.**Problem: , proposed by

**Ah**

**met Arduc**, Tip: Key Fact(s):

**B13A**

64 is the smallest number with 7 divisors.

**65.**Problem: , proposed by

**Ah**

**met Arduc**

65 is the lowest integer that becomes square if its reverse is either added to or subtracted from it.

**66.**Problem: , proposed by Amirul Faiz Abdul Muthalib

66 is a triangular palindromic number.

**67.**Problem: , proposed by

**Ah**

**met Arduc**

67 is the smallest number which is both palindromic in base 5 and in base 6.

**68.**Problem: , proposed by Amirul Faiz Abdul Muthalib

68 is the smallest composite number that becomes prime by turning it upside down.

**69.**Problem: , proposed by

**Ah**

**met Arduc**

69 is a value of n where n² and n³ together contain each digit once.

**72.**Problem: , proposed by

**Ah**

**met Arduc**

72 is the maximum number of spheres that can touch another sphere in a lattice packing in 6 dimensions.

**73.**Problem: , proposed by Ihsan Yucel

73 is the smallest multi-digit number which is one less than twice its reverse.

**74.**Problem: , proposed by Ihsan Yucel

74 is the number of non-Hamiltonian polyhedra with a minimum number of vertices.

**77.**Problem: , proposed by

**Ah**

**met Arduc**

77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals add up to 1.

**78.**Problem: , proposed by Ihsan Yucel

78 is the lowest number which can be written as the sum of 4 distinct squares in 3 ways.

**80.**Problem: , proposed by

**Ah**

**met Arduc**

80 is the smallest number with exactly 7 representations as a sum of three distinct primes.

**81.**Problem: , proposed by

**Ah**

**met Arduc**

81 is the square of the sum of its digits; there are no other numbers with this property except 0 and 1.

**84.**Problem: , proposed by

**Ah**

**met Arduc**

84 is the smallest integer with 8 different representations as a sum of 2 primes.

**88.**Problem: , proposed by

**Ah**

**met Arduc**

88 can be read the same upside down or when viewed in a mirror.

**90.**Problem: , proposed by

**Ah**

**met Arduc**

90 is the smallest number having 6 representations as a sum of four positive squares.

**92.**Problem: , proposed by

**Ah**

**met Arduc**

92 is the number of different arrangements of 8 non-attacking queens on an 8x8 chessboard.