# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1026: Dualities

From New Scientist #2182, 17th April 1999 [link]

Across:

1. A prime which is also a square reversed. The first two digits form a square, and the last two a prime. The 1st, 3rd and 5th digits are all the same.
4. The square root of 7 across.
5. A palindromic square.
6. The square root of the reverse of 2 down.
7. A square which is prime when reversed.

Down:

1. A prime which is also a square reversed. The first three digits form a square which is also a square when reversed. The last two digits form a prime which is also a prime when reversed.
2. A prime which is also a square when reversed. All the digits are different. The first three digits form a square which is also a square when reversed: and the last digit is the same as that of 1 down.
3. A square which is a prime when reversed.

Find the answers for 1 across, 1 down, 3 down and 7 across.

[enigma1026]

### One response to “Enigma 1026: Dualities”

1. Jim Randell 25 January 2019 at 7:52 am

Several of the digits are equated with each other, either directly or by the use of palindromic numbers.

We can apply these and then use the [[ `SubstitutedExpression()` ]] solver from the enigma.py library to solve the remaining conditions of the puzzle.

The following run file executes in 142ms.

Run: [ @repl.it ]

```#  we transform the first grid to the second by equating:
#
#  A = C = E, P = K, N = L, X = V
#
#  A B C D E      A B A D A
#  F G H I J      F G H I J
#  K L M N P  ->  K L M L K
#  Q R S T U      Q R S T U
#  V W X Y Z      V W V Y Z

SubstitutedExpression

--distinct="AHMSV"
--template="Across: 1. {ABADA}; 4. {GHI}; 5. {KLMLK}; 6. {RST}; 7. {VWVYZ} / Down: 1. {AFKQV}; 2. {AHMSV}; 3. {AJKUZ}"
--solution=""

# "A prime ..."
# "... which is also a square reversed ..."
# "... the first 2 digits form a square ..."
"is_square(AB)"
# "... the last 2 digits form a prime ..."
"is_prime(DA)"

# 4a = GHI
# "the square root of 7 across"
"GHI ** 2 = VWVYZ"

# 5a = KLMLK
# "a palindromic square"
"is_square(KLMLK)"

# 6a = RST
# "the square root of the reverse of 2 down"
"RST ** 2 = VSMHA"

# 7a = VWVYZ
# "a square ..."
"is_square(VWVYZ)"
# "... which is a prime when reversed"
"is_prime(ZYVWV)"

# 1d = AFKQV
# "a prime ..."
"is_prime(AFKQV)"
# "... which is also a square reversed ..."
"is_square(VQKFA)"
# "... the first 3 digits form a square ..."
"is_square(AFK)"
# "... which is also a square when reversed ..."
"is_square(KFA)"
# "... the last 2 digits form a prime ..."
"is_prime(QV)"
# "... which is also a prime when reversed"
"is_prime(VQ)"

# 2d = AHMSV
# "a prime ..."
"is_prime(AHMSV)"
# "... which is also a square when reversed ..."
"is_square(VSMHA)"
# "... all the digits are different ..." (see --distinct)
# "... the first 3 digits form a square ..."
"is_square(AHM)"
# "... which is also a square when reversed ..."
"is_square(MHA)"

# 3d = AJKUZ
# "a square ..."
"is_square(AJKUZ)"
# "... which is a prime when reversed"
"is_prime(ZUKJA)"
```

Solution: 1 across = 16141; 1 down = 14437; 3 down = 18496; 7 across = 70756.

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