### Solution to Problem #5

The problem was solved by Joel Haywood of Macon GA, Richard Roland of Memphis TN, Darryl K. Nester of Bluffton College, John Shonder of Oak Ridge TN, James D. Beltz of St. Charles MO, Matt Hudelson of Washington State University, Robin Stokes of the University of New England (Australia), Tye Rattenbury of the University of Colorado, Ross Millikan of San Mateo CA, Jason Curtright of Wentzville MO, Dwayne Hoffman of Apollo Beach FL, Ryan Moats of Omaha NE, Michael Byrd of Southwest Missouri State University, Philippe Fondanaiche of Paris (France), Carlos Rivera of Nuevo Leon (Mexico), Justin Sorice of Bluffton College, Josie Madl of Southwest Missouri State University, David Snook of Port Coquitlam (British Columbia), Glenn Lamb of Mountain View CA, Luke McCarthy of Queen's University, Theo Koupelis of the University of Wisconsin - Marathon, Mario Roederer of Stanford University, Jakkidi Srinivas of Southwest Missouri State University, Ken Duisenberg of Roseville CA, Susan Hoover of Houston TX, Christina Tonsing of Springfield MO, and Caleb Lenz and Andrew Underwood both from Glendale High School (Springfield MO). Robin Stokes, Ryan Moats, and Susan Hoover also found solutions in other number bases. Here is Darryl Nester's solution followed by the results of Robin Stokes, Ryan Moats, and Susan Hoover:

```Answer: O=2, N=3, E=6, T=4, W=7, S=5, V=9, I=1

... so that ONE + ONE = TWO is 236 + 236 = 472,
the prime number SEVEN is 56963,
and the perfect square NINE is 3136=56^2.

Solution: By scanning the squares of the number 32 to 99, we find that there
are only 4 four-digit squares which match the pattern NINE -- that is, only
4 of these squares have the same first and third digits.  These are 2025,
3136, 6561, and 8281.  This means that N is one of 2, 3, 6, or 8, but since
SEVEN is to be prime, N must be odd.  Thus, N=3, I=1, and E=6.

Then immediately we have O=2, since 2*E must end in the digit O ("oh"), so
that W=7 and T=4.

Finally, the number SEVEN is _6_63, with the first blank chosen from 5, 8,
or 9 (the only remaining nonzero digits), and the second blank chosen from
0, 5, 8, or 9.  Searching a table of primes can reveal which of these 9
possible numbers is prime; alternatively, use a TI-92 and the factor command
(or corresponding methods on a similar tool):

factor(56063+{0,800,900})
reveals that 56963 is prime; the other two are composite;
factor(86063+{0,500,900})
reveals that all three are composite;
factor(96063+{0,500,800})
reveals that all three are composite.

(If leading zeros were allowed, we could have S=0 and V=5 or 8, since both
06563 and 06583 are prime.)
```

There are no solutions base 8, 9, 11, 12, 14, 15, or 20. Here are the solutions in bases 10, 13, and 16 through 19 (a=10, b=11, etc.):

```		one	two	nine	seven

base 10:	236	472	3136	56963
base 13:	589	b45	8089	29398
base 13:	589	b45	8089	29798
base 13:	589	b45	8089	39c98
base 13:	589	b45	8089	79298
base 16:	231	462	3931	51d13
base 16:	231	462	3931	71e13
base 16:	231	462	3931	81013
base 16:	231	462	3931	81d13
base 16:	231	462	3931	a1713
base 16:	231	462	3931	b1f13
base 16:	231	462	3931	c1013
base 16:	231	462	3931	c1513
base 16:	231	462	3931	d1b13
base 16:	231	462	3931	d1e13
base 16:	231	462	3931	e1713
base 16:	231	462	3931	f1b13
base 17:	139	271	3a39	49g93
base 17:	139	271	3a39	d9f93
base 17:	139	271	3a39	f9b93
base 17:	1c9	381	cfc9	2959c
base 17:	1c9	381	cfc9	29b9c
base 17:	1c9	381	cfc9	59e9c
base 17:	1c9	381	cfc9	69b9c
base 17:	1c9	381	cfc9	7909c
base 17:	1c9	381	cfc9	79g9c
base 17:	1c9	381	cfc9	b929c
base 17:	1c9	381	cfc9	e959c
base 17:	462	8c4	6b62	12g26
base 17:	462	8c4	6b62	72a26
base 17:	462	8c4	6b62	e2326
base 17:	462	8c4	6b62	e2726
base 17:	462	8c4	6b62	e2d26
base 17:	462	8c4	6b62	g2126
base 17:	4e2	9b4	e6e2	1282e
base 17:	4e2	9b4	e6e2	5282e
base 17:	4e2	9b4	e6e2	7202e
base 17:	4e2	9b4	e6e2	8252e
base 17:	4e2	9b4	e6e2	a232e
base 17:	4e2	9b4	e6e2	a252e
base 17:	4e2	9b4	e6e2	a2f2e
base 17:	4e2	9b4	e6e2	c232e
base 17:	4e2	9b4	e6e2	c272e
base 17:	4e2	9b4	e6e2	f2c2e
base 17:	4e2	9b4	e6e2	g232e
base 17:	864	gc8	6064	24746
base 17:	864	gc8	6064	24b46
base 17:	864	gc8	6064	54246
base 17:	864	gc8	6064	54a46
base 17:	864	gc8	6064	a4546
base 17:	864	gc8	6064	e4b46
base 18:	25a	4b2	515a	3aha5
base 18:	25a	4b2	515a	6a7a5
base 18:	25a	4b2	515a	6a8a5
base 18:	25a	4b2	515a	6aca5
base 18:	25a	4b2	515a	7a0a5
base 18:	25a	4b2	515a	7ada5
base 18:	25a	4b2	515a	7afa5
base 18:	25a	4b2	515a	7aga5
base 18:	25a	4b2	515a	8aca5
base 18:	25a	4b2	515a	8aea5
base 18:	25a	4b2	515a	ca6a5
base 18:	25a	4b2	515a	da0a5
base 18:	25a	4b2	515a	da9a5
base 18:	25a	4b2	515a	eafa5
base 18:	25a	4b2	515a	fa3a5
base 18:	25a	4b2	515a	fada5
base 18:	25a	4b2	515a	faha5
base 18:	25a	4b2	515a	ga9a5
base 18:	25a	4b2	515a	gaha5
base 18:	25a	4b2	515a	hada5
base 18:	25a	4b2	515a	haga5
base 19:	2b1	532	bhb1	4161b
base 19:	2b1	532	bhb1	41e1b
base 19:	2b1	532	bhb1	41i1b
base 19:	2b1	532	bhb1	71f1b
base 19:	2b1	532	bhb1	8141b
base 19:	2b1	532	bhb1	81a1b
base 19:	2b1	532	bhb1	81g1b
base 19:	2b1	532	bhb1	91d1b
base 19:	2b1	532	bhb1	c1g1b
base 19:	2b1	532	bhb1	e1a1b
base 19:	2b1	532	bhb1	e1g1b
base 19:	2b1	532	bhb1	f1d1b
base 19:	2b1	532	bhb1	g1c1b
base 19:	2b1	532	bhb1	i141b
base 19:	2d1	572	dad1	6141d
base 19:	2d1	572	dad1	61g1d
base 19:	2d1	572	dad1	8101d
base 19:	2d1	572	dad1	b1h1d
base 19:	2d1	572	dad1	c141d
base 19:	2d1	572	dad1	e161d
base 19:	2d1	572	dad1	h1b1d
base 19:	2d1	572	dad1	h1f1d
base 19:	2d1	572	dad1	i181d
base 19:	3gb	7e3	g6gb	4bdbg
base 19:	3gb	7e3	g6gb	5b4bg
base 19:	3gb	7e3	g6gb	5babg
base 19:	3gb	7e3	g6gb	5bcbg
base 19:	3gb	7e3	g6gb	9b0bg
base 19:	3gb	7e3	g6gb	fb0bg
base 19:	3gb	7e3	g6gb	fb8bg
base 19:	3gb	7e3	g6gb	fbibg
base 19:	3gb	7e3	g6gb	hb0bg
base 19:	3gb	7e3	g6gb	ibhbg
base 19:	814	g28	1714	64a41
base 19:	814	g28	1714	64e41
base 19:	814	g28	1714	94h41
base 19:	814	g28	1714	b4341
base 19:	814	g28	1714	b4941
base 19:	814	g28	1714	b4f41
base 19:	814	g28	1714	d4f41
base 19:	814	g28	1714	e4c41
base 19:	814	g28	1714	f4541
base 19:	814	g28	1714	h4941
base 19:	864	gc8	6964	24346
base 19:	864	gc8	6964	54046
base 19:	864	gc8	6964	54a46
base 19:	864	gc8	6964	54i46
base 19:	864	gc8	6964	d4a46
base 19:	864	gc8	6964	e4146
base 19:	864	gc8	6964	i4b46
base 19:	864	gc8	6964	i4f46
```

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