Six deceptive problems that no one can solve

This is the title of the article which appeared here 

The problems are
1. Twin Prime Conjecture
2. The Moving Sofa Problem
3. The Collatz Conjecture
4. The Beal Conjecture
5. The Inscribed Square Problem
6. Goldbach Conjecture

I am aware of the problems number 1,3 and 6. So I need to investigate the other 3. Which are Moving Sofa, Beal Conjecture and the Inscribed Square Problem.

How many triangular numbers are also Fibonacci Numbers

I heard this from Bruce Edwards that they are only 5 and these are 1,1,3,21, and 55. I wrote a program in Sagemath to verify that. Here it is Here is the program with its output

def fibo(n):
    fib_1 = 1
    fib_2 = 1
    if n == 1 or n== 2:
        return 1
    fib_n=0
   
    for i in range(3,n+1):
        fib_n = fib_1+fib_2
        fib_1 = fib_2
        fib_2 = fib_n
    return fib_n
def triNumList(n):
    a=[]
    for i in range(1,n):
        b=i*(i+1)/2
        a.append(b)
    return a

def fiboList(n):
    a=[]
    for i in range(1,n):
        a.append(fibo(i))
    return a

def TriFiboCompare(n):
    f = fiboList(n)
    t = triNumList(n)
    print(f)
    print(t)
    for i in f:
        if i in t:
            print("Match",i)

TriFiboCompare(100)


[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 19740274219868223167, 31940434634990099905, 51680708854858323072, 83621143489848422977, 135301852344706746049, 218922995834555169026] [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2628, 2701, 2775, 2850, 2926, 3003, 3081, 3160, 3240, 3321, 3403, 3486, 3570, 3655, 3741, 3828, 3916, 4005, 4095, 4186, 4278, 4371, 4465, 4560, 4656, 4753, 4851, 4950] ('Match', 1) ('Match', 1) ('Match', 3) ('Match', 21) ('Match', 55)

Tiling a deleted 64 grid using dominos

This is a classic problem that can be found in most books on problem solving and I re-encountered on my reading of Mathematics a Very Short Introduction by Timothy Gower. The problem is if you delete the two diagonal squares of a 64 grid squares like a chess board and then you have to tile it using dominos. The way to do is to think of is to realize that each domino occupies a black and white square. Assuming that the two black squares were deleted. There are now 62 squares in total out of which 32 are white and 30 are black. So no matter how we tile. We will be left with 2 white squares and since a domino tiles only black and white those two will remain untiled.