Solution to the Build Your Own Sudoku puzzle

Build Your Own Sudoku is one of the interesting puzzles of the MIT Mystery Hunt 2014 competition. The plot of the game that year was based on Lewis Carroll's "Alice's Adventures in Wonderland" and "Through the Looking-Glass and what Alice found there." The puzzle featured a blank Sudoku grid with four sets of clues. The first three sets were needed to fill in the puzzle's initial numbers: they corresponded to the sectors, rows and columns of the grid respectively. The last set corresponded to the grid's numbers and was needed to get the answer to the puzzle after solving the Sudoku. This puzzle was designed by Robbie Buckingham.

	1	2	3	4	5	6	7	8	9
A
B
C
D
E
F
G
H
I

Sectors

Northwest	No square with a given number in the Northwest Sector touches another square with a given number in the same sector along an edge. (In general we say a square touches another if it is the same square or it shares an edge or corner.)
North	The sum of the given numbers in the North Sector is 13.
Northeast	There is a continuous path of touching squares with given numbers from the upper-right corner of the sudoku grid to the lower-left corner of the sudoku grid.
West	The sum of the given numbers in the West Sector is 27, while their product is 1512.
Center	Rotational symmetry: if there is a given number in any square of the sudoku grid, there is also a (possibly different) given number in the square 180° opposite.
East	The product of the given numbers in the East Sector is 288.
Southwest	Square I1 matches its antipodal square (A9) in the unsolved grid.
South	The sum of the given numbers in the South Sector is 16.
Southeast	The given numbers in the Southeast Sector are exactly those in the East Sector.

Rows

A	The third square of Row A (A3) matches its antipodal square (I7) in the unsolved grid.
B	The given numbers in Row B are exactly those in Column 6.
C	The leftmost given number in Row C divides all the other given numbers in its sector.
D	The product of the given numbers in D8 and D9 is 16.
E	None of the squares with given numbers in Row E are directly above any of the four squares with given numbers in the row below, or directly below any of the squares with given numbers in the row above.
F	The sum of the given numbers in Row F, which is the same as the sum of the given numbers in Row D, is 20.
G	Row G, and every other row, has at least three given numbers.
H	The product of the given numbers in H3 and H4 is 72.
I	The sum of any given numbers in I1, I2, I3, and I4 is 6 more than the sum of any given numbers in H1 and H2, but 3 less than the sum of any given numbers in Column 5.

Columns

1	The product of any given numbers in D1, E1, F1, and G1 is 63.
2	The product of the given numbers in Column 2 is 18 more than the product of any given numbers in the Northeast Sector.
3	The sum of the given numbers in Column 3 is 17.
4	The three squares with given numbers in Column 4 are touching.
5	There is only one given number in Column 5.
6	There is only one square with a given number touching G6.
7	None of the four squares with given numbers in Column 7 touch each other.
8	There are the same number of squares with given numbers touching A8 as there are touching B8.
9	The product of the given numbers in Column 9 is 180.

Numbers

1	=	A1	×	B7	+	C5	−	?
2	=	B2	×	C5	+	D2	−	?
3	=	C3	×	D7	+	E3	−	?
4	=	D4	×	E9	+	F3	−	?
5	=	E5	×	F4	+	G7	−	?
6	=	F6	×	G1	+	H5	−	?
7	=	G7	×	H9	+	I5	−	?
8	=	H8	×	I3	+	A3	−	?
9	=	I9	×	A9	+	B1	−	?

Solution

We will mark the squares of the Sudoku grid that must contain a number using the symbol '*' and the squares that must remain empty using the symbol 'x'.

	1	2	3	4	5	6	7	8	9
A
B
C
D
E
F
G
H
I

According to the clues for the Column 5 and the Center Sector this column will contain only one number in the middle, in the square E5.

	1	2	3	4	5	6	7	8	9
A					х
B					x
C					x
D					x
E					*
F					x
G					x
H					x
I					x

Let's consider the clue for Column 8, taking into account the definition of touching of squares given in the clue for the Northwest Sector. In the Northeast Sector the squares A7, A8, A9, B7, B8, B9 touch both the squares A8 and B8; and the squares C7, C8, C9 touch only the square B8. Thus, numbers can be in any of the squares A7, A8, A9, B7, B8, B9, but cannot be in the squares C7, C8, C9. Symmetrically (clue for the Center Sector), the squares G1, G2, G3 will be empty.

	1	2	3	4	5	6	7	8	9
A					х
B					x
C					x		x	x	x
D					x
E					*
F					x
G	x	x	x		x
H					x
I					x

According to the clue for the Northeast Sector the squares A9, C6, G4, I1 will be filled.

	1	2	3	4	5	6	7	8	9
A					х				*
B					x
C					x	*	x	x	x
D					x
E					*
F					x
G	x	x	x	*	x
H					x
I	*				x

According to the clue for the Row H the squares H3 and H4 will be filled. Symmetrically, the squares B6 and B7 will be filled.

	1	2	3	4	5	6	7	8	9
A					х				*
B					x	*	*
C					x	*	x	x	x
D					x
E					*
F					x
G	x	x	x	*	x
H			*	*	x
I	*				x

According to the clue for the Column 4 the squares A4, B4, C4, D4, E4 will be empty. Symmetrically, the squares E6, F6, G6, H6, I6 will be empty.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D				x	x
E				x	*	x
F					x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Row D the squares D8 and D9 must contain numbers 2 and 8. According to the clue for the Column 9 there cannot be an 8 in that column, since 180 is not divisible by 8. In this case the square D8 will contain the number 8, and the square D9 will contain the number 2. Symmetrically, the squares F1 and F2 will be filled.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D				x	x			8	2
E				x	*	x
F	*	*			x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Row E the squares E1, E2, E8, E9 will be empty.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D				x	x			8	2
E	x	x		x	*	x		x	x
F	*	*			x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Row G the squares E3 and E7 will be filled.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D				x	x			8	2
E	x	x	*	x	*	x	*	x	x
F	*	*			x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Row E the squares D3, D7, F3, F7 will be empty.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D			x	x	x		x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x		x	x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Northeast Sector the squares D6 and F4 will be filled.

	1	2	3	4	5	6	7	8	9
A				x	х				*
B				x	x	*	*
C				x	x	*	x	x	x
D			x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*				x	x

According to the clue for the Column 4 the square I4 will be empty. Symmetrically, the square A6 will be empty.

	1	2	3	4	5	6	7	8	9
A				x	х	x			*
B				x	x	*	*
C				x	x	*	x	x	x
D			x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x
G	x	x	x	*	x	x
H			*	*	x	x
I	*			x	x	x

According to the clue for the Row H the squares H3 and H4 must contain the numbers 8 and 9. According to the clue for the South Sector the sum of the numbers in the squares G4 and H4 must equal 16. If we assume that the square H4 contains the number 8, then the square G4 must also contain the number 8, which is contrary to the rules of Sudoku. Therefore, the square H3 will contain the number 8, the square H4 will contain the number 9 and the square G4 will contain the number 7.

	1	2	3	4	5	6	7	8	9
A				x	х	x			*
B				x	x	*	*
C				x	x	*	x	x	x
D			x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x
G	x	x	x	7	x	x
H			8	9	x	x
I	*			x	x	x

According to the clue for the Column 1 the square D1 will be filled. Symmetrically, the square F9 will be filled.

	1	2	3	4	5	6	7	8	9
A				x	х	x			*
B				x	x	*	*
C				x	x	*	x	x	x
D	*		x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x		*
G	x	x	x	7	x	x
H			8	9	x	x
I	*			x	x	x

According to the clue for the Row E the Row F must have four squares filled in. Therefore, the square F8 will be empty. Symmetrically, the square D2 will be empty.

	1	2	3	4	5	6	7	8	9
A				x	х	x			*
B				x	x	*	*
C				x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x	x	*
G	x	x	x	7	x	x
H			8	9	x	x
I	*			x	x	x

According to the clue for the Column 7 the squares A7 and H7 will be empty and the squares G7 and I7 will be filled. Symmetrically, the squares B3 and I3 will be empty and the squares A3 and C3 will be filled.

	1	2	3	4	5	6	7	8	9
A			*	x	х	x	x		*
B			x	x	x	*	*
C			*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x	x	*
G	x	x	x	7	x	x	*
H			8	9	x	x	x
I	*		x	x	x	x	*

According to the clue for the Northwest Sector the squares A2 and C2 will be empty. Symmetrically, the squares G8 and I8 will be empty.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B			x	x	x	*	*
C		x	*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x	x	*
G	x	x	x	7	x	x	*	x
H			8	9	x	x	x
I	*		x	x	x	x	*	x

According to the clue for the Row G the squares C1 and G9 will be filled.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B			x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x	x	*
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x
I	*		x	x	x	x	*	x

According to the clue for the Northwest Sector the square B1 will be empty. Symmetrically, the square H9 will be empty.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	*	x	*	x	*	x	x
F	*	*	x	*	x	x	x	x	*
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	*	x

According to the clue for the Column 1 the squares D1 and F1 of the West Sector must contain the numbers 7 and 9. Let's consider the clue for the West Sector, according to which the sum of the numbers in the squares E3 and F2 is 27 - (7 + 9) = 11, and their product is 1512/63 = 24. Thus, these squares must contain the numbers 3 and 8. Since the Column 3 already contains the number 8, the square E3 will contain the number 3, and the square F2 will contain the number 8.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	3	x	*	x	*	x	x
F	*	8	x	*	x	x	x	x	*
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	*	x

According to the clue for the East Sector the product of the numbers in the squares E7 and F9 is 288/(2*8) = 18. In this case these squares can contain the numbers 2 and 9 or 3 and 6. However, the East Sector already contains the number 2, so only the numbers 3 and 6 remain. Since the Row E already contains the number 3, the square E7 will contain the number 6, and the square F9 will contain the number 3.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	*	x	x	x	x	*	x	8	2
E	x	x	3	x	*	x	6	x	x
F	*	8	x	*	x	x	x	x	3
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	*	x

According to the clue for the Column 1 the squares D1 and F1 must contain the numbers 7 and 9. Let's suppose the square F1 contains the number 9. In this case, according to the clue for the Row F, the square F4 will contain 20 - (9 + 8 + 3) = 0, which is contrary to the rules of Sudoku. Therefore, the square D1 will contain the number 9 and the square F1 will contain the number 7.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	9	x	x	x	x	*	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	*	x	x	x	x	3
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	*	x

According to the clue for the Row F the square D6 will contain 20 - (9 + 8 +2) = 1, and the square F4 will contain 20 - (7 + 8 + 3) = 2.

	1	2	3	4	5	6	7	8	9
A		x	*	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	*	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	*	x

According to the clue for the Column 3 the sum of the numbers in the squares A3 and C3 is 17 - (3 + 8) = 6. Thus, these squares can contain the numbers 1 and 5 or 2 and 4. Then, according to the clue for the Row A, the squares A3 and I7 must match. In turn, according to the clue for the Southeast Sector, the numbers in this sector must be exactly the same as in the East Sector, namely: 2, 3, 6, 8. Thus, among the numbers 1, 2, 4, 5, the only possible option for the squares A3 and I7 is the number 2. Therefore, the square C3 will contain the number 4.

	1	2	3	4	5	6	7	8	9
A		x	2	x	х	x	x		*
B	х		x	x	x	*	*
C	*	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	2	x

According to the clue for the Row C and the numbers already in the Northwest Sector the square C1 will contain the number 1.

	1	2	3	4	5	6	7	8	9
A		x	2	x	х	x	x		*
B	х		x	x	x	*	*
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	*
H			8	9	x	x	x		х
I	*		x	x	x	x	2	x

Let's consider the square G9, which must be filled. According to the clue for the Southeast Sector it must contain one of the numbers of the East Sector, namely: 2, 3, 6, 8. Since the Column 9 already contains the numbers 2 and 3, that leaves only the numbers 6 and 8. According to the clue for the Column 9 this column cannot contain the number 8, since 180 is not divisible by 8. Therefore, the square G9 will contain the number 6.

	1	2	3	4	5	6	7	8	9
A		x	2	x	х	x	x		*
B	х		x	x	x	*	*
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	6
H			8	9	x	x	x		х
I	*		x	x	x	x	2	x

According to the clue for the Column 9 the product of its remaining numbers is 180/(2*3*6) = 5. Therefore, this column must contain the number 5, and it can also contain the number 1. According to the clue for the Southeast Sector neither of these numbers can be in the square I9, which will therefore be empty. Symmetrically, the square A1 will be empty.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x		*
B	х		x	x	x	*	*
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	6
H			8	9	x	x	x		х
I	*		x	x	x	x	2	x	х

Let's now consider the square A9, which, as shown above, can contain the numbers 1 and 5. Let's assume it contains the number 1. Then, according to the clue for the Southwest Sector, the square I1 must also contain the number 1; however, the Column 1 already contains the number 1, which is contrary to the rules of Sudoku. Therefore, the squares A9 and I1 will contain the number 5.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x		5
B	х		x	x	x	*	*
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	*	x	6
H			8	9	x	x	x		х
I	5		x	x	x	x	2	x	х

According to the clue for the Southeast Sector the squares G7 and H8 must contain the numbers 3 and 8. Since the Row H already contains the number 8, the square G7 will contain the number 8, and the square H8 will contain the number 3. Symmetrically, the square B2 will be filled.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x		5
B	х	*	x	x	x	*	*
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H			8	9	x	x	x	3	х
I	5		x	x	x	x	2	x	х

According to the clue for the Row B this row must contain the same number of given numbers as the Column 6, namely three. Therefore, the squares B8 and B9 will be empty. Symmetrically, the squares H1 and H2 will be empty.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x		5
B	х	*	x	x	x	*	*	х	х
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5		x	x	x	x	2	x	х

According to the clue for the Northeast Sector the squares A8 and I2 will be filled.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	*	5
B	х	*	x	x	x	*	*	х	х
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	*	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	*	x	x	x	x	2	x	х

Let's consider the clue for the Row I. The squares H1 and H2 will be empty, so we'll assume their corresponding sum is zero. In this case the square I2 will contain the number 1, which means the square E5 will contain the number 9.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	*	5
B	х	*	x	x	x	*	*	х	х
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	9	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	1	x	x	x	x	2	x	х

According to the clue for the Row B the square B7 will contain the number 1.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	*	5
B	х	*	x	x	x	*	1	х	х
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	9	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	1	x	x	x	x	2	x	х

According to the clue for the Column 2 we have: B2*8*1 - 18 = A8*5*1 or 8*B2 - 18 = 5*A8. From which we obtain: B2 = 6 and A8 = 6.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	6	5
B	х	6	x	x	x	*	1	х	х
C	1	x	4	x	x	*	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	9	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	1	x	x	x	x	2	x	х

According to the clue for the Row B the square C6 will contain the number 6.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	6	5
B	х	6	x	x	x	*	1	х	х
C	1	x	4	x	x	6	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	9	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	1	x	x	x	x	2	x	х

According to the clue for the North Sector the square B6 will contain the number 7.

	1	2	3	4	5	6	7	8	9
A	х	x	2	x	х	x	x	6	5
B	х	6	x	x	x	7	1	х	х
C	1	x	4	x	x	6	x	x	x
D	9	x	x	x	x	1	x	8	2
E	x	x	3	x	9	x	6	x	x
F	7	8	x	2	x	x	x	x	3
G	x	x	x	7	x	x	8	x	6
H	х	х	8	9	x	x	x	3	х
I	5	1	x	x	x	x	2	x	х

Thus, we finally obtain the unsolved Sudoku grid.

	1	2	3	4	5	6	7	8	9
A			2					6	5
B		6				7	1
C	1		4			6
D	9					1		8	2
E			3		9		6
F	7	8		2					3
G				7			8		6
H			8	9				3
I	5	1					2

The next step is to actually solve the Sudoku.

	1	2	3	4	5	6	7	8	9
A	8	7	2	1	3	9	4	6	5
B	3	6	5	8	4	7	1	2	9
C	1	9	4	5	2	6	3	7	8
D	9	4	6	3	7	1	5	8	2
E	2	5	3	4	9	8	6	1	7
F	7	8	1	2	6	5	9	4	3
G	4	3	9	7	1	2	8	5	6
H	6	2	8	9	5	4	7	3	1
I	5	1	7	6	8	3	2	9	4

And now we should use the final set of clues for the numbers in the solved Sudoku grid. The result is a sequence of numbers: 9-14-20-18-21-19-9-15-14. Converting these numbers into letters yields the final answer, INTRUSION.