A344331 Side s of squares of type 1 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

10, 20, 30, 40, 50, 60, 68, 70, 78, 80, 90, 100, 110, 120, 130, 136, 140, 150, 156, 160, 170, 180, 190, 200, 204, 210, 220, 222, 230, 234, 240, 250, 260, 270, 272, 280, 290, 300, 310, 312, 320, 330, 340, 350, 360, 370, 380, 390, 400, 408, 410, 420, 430, 440, 444, 450, 460, 468, 470

Offset: 1

Author

Bernard Schott, May 20 2021

Keywords

• nonn

Comments

This sequence is relative to the generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008 (see A344330).

There are two types of solutions, the first one is proposed here, while type 2 is described in A344332.

Some notations: s = side of the tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.

-> Primitive squares

Side s of primitive squares of type 1 must satisfy the Diophantine equation s^2 = z * (a^2+b^2), with gcd(a, b) = 1, and without using the conditions a^2+b^2 = c^2, when a and b belong to a Pythagorean triple (a, b, c).

In this case, the sides of the primitive squares of type 1 are s = a*b * (a^2+b^2) with 1 <= a < b and gcd(a, b) = 1 (A344333), then corresponding z = (a*b)^2 * (a^2+b^2) (A344334).

Every primitive square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.

In particular: for a = 1, b = n, s = n*(n^2+1) form the subsequence A034262 \ {0, 1} and z = n^2*(n^2+1) form the subsequence A071253 \ {0, 2}).

See example with design for a square of side s = 10 with a = 1, b = 2, m = 10, z = 20.

-> Non-primitive squares

If s is the side of a primitive square of type 1 with z squares of side a and z squares of side b, then every k * s is a non-primitive term that gives one or two distinct tilings of type 1, depending of value of k:

- For every k > 1, the square ks X ks can be tiled with k^2*z squares of side a and k^2*z squares of side b (see example).

- For every k = r^4, r>1, the square ks X ks also can be tiled with z squares of side ka and z squares of side kb.

---> Consequences:

1) For every pair (a, b), 1 <= a < b, there is a square of side s = a*b * (a^2+b^2) that can be tiled with squares of side a and side b so that the number z of squares of side a and side b is the same, this number z = (a*b)^2 * (a^2+b^2).

2) If q is a term and K > 1, K * q is another term.

3) Every term is even.

Examples

			Primitive square with s = 10:
   a = 1, b = 2, s = 10, m = 10, z = 20, and
Non-primitive square with s = 20:
   a = 1, b = 2, s = 20, m = 40, z = 80.
      ___ ___ _ ___ ___ _ ___________________
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|___________________|
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |___________________|___________________|
with respectively m = 10 (and m = 40) elementary 2 X 5 rectangles as below:
          ___ ___ _
         |   |   |_|
         |___|___|_|
There are these three possibilities:
- 10 is a primitive term because the square 10 X 10 can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 2 X 2.
- 20 is a non-primitive term because the square 20 X 20 can be tiled with 80 squares of size 1 X 1 and 80 squares of size 2 X 2.
- 30 is a primitive term because the square 30 X 30 can be tiled with 90 squares of size 1 X 1 and 90 squares of size 3 X 3, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 3 X 3,
  but also, 30 is a non-primitive term because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2.
		

References

• Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Links

• Index to sequences related to Olympiads.

Crossrefs

Cf. A344330, A344332, A344333, A344334.

Cf. A034262, A071253.

Programs

PARI

isokp1(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->x*y*(x^2+y^2), [1..m]), s); }
isok(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Dec 22 2021