in the cryptarithm shown, determine the greatest possible value of SIGN? PEN + INK = SIGN

Question

In a cryptarithm, every letter represents a different digit.

in the cryptarithm shown, determine the greatest possible value of SIGN?

 

PEN + INK = SIGN

Answer 1

Let’s write the cryptarithm this way:
      P   E   N
  +  I   N   K
= = = = = = =
  S  I   G   N

For the cryptarithm to make sense, S ≠ 0.
Also, S ≤ 2 [since y ≤ 9, P ≤ 9 and I ≤ 9]

To solve, we could (1) use Exhaustive Enumeration [easy, using a computer] or (2) systematically apply rules of math.

Using method (1) [note: I did it – comment: low-to-high values make logic simple and computers are very, very fast],
FOR S = 1 TO 2
  FOR I = 0 TO 9
    FOR G = 0 TO 9
      FOR N = 0 TO 9
        [and for values of P, E, and K]
        IF S,I,G,N,P,E,K are unique AND IF PEN+INK=SIGN THEN
          OUTPUT (P*100+E*10+N) + " + " + (I*100+N*10+K) + " = " + (S*1000+I*100+G*10+N)
      NEXT N
    NEXT G
  NEXT I
NEXT S

the result is:
985 + 250 = 1235
976 + 260 = 1236
967 + 270 = 1237
958 + 280 = 1238
986 + 260 = 1246
968 + 280 = 1248
987 + 270 = 1257
978 + 280 = 1258
984 + 340 = 1324
975 + 350 = 1325
957 + 370 = 1327
948 + 380 = 1328
986 + 360 = 1346
968 + 380 = 1348
987 + 370 = 1357
978 + 380 = 1358
975 + 450 = 1425
957 + 470 = 1427
985 + 450 = 1435
976 + 460 = 1436
967 + 470 = 1437
958 + 480 = 1438
987 + 470 = 1457
978 + 480 = 1458
984 + 540 = 1524
948 + 580 = 1528
976 + 560 = 1536
967 + 570 = 1537
986 + 560 = 1546
968 + 580 = 1548
984 + 640 = 1624
975 + 650 = 1625
957 + 670 = 1627
948 + 680 = 1628
985 + 650 = 1635
958 + 680 = 1638
987 + 670 = 1657
978 + 680 = 1658
984 + 740 = 1724
948 + 780 = 1728
985 + 750 = 1735
958 + 780 = 1738
986 + 760 = 1746
968 + 780 = 1748
975 + 850 = 1825
957 + 870 = 1827
976 + 860 = 1836
967 + 870 = 1837

These are ALL the values that satisfy the criteria. Simply choose the last line (it is the largest possible value of SIGN).

Using method (2), here is what we know:
      P   E   N
  +  I   N   K
= = = = = = =
  S  I   G   N

Now, in adding three-digit numbers, there are three “carries.” Let’s call them x, y and z (right-to-left).
N + K = 10x + N         [that is, digits “xN”]
x + E + N = 10y + G
y + P + I = 10z + I

For the cryptarithm to make sense, S≠ 0.
Also, S ≤ 2 [since y ≤ 9, P ≤ 9 and I ≤ 9]
and x ≤ 2 and y ≤ 2

S = z        [S is simply “putting down the carry z”]

Can S = 2 (that makes SIGN the greatest) ?
That means that (P+y) must equal 10 (since y+P+I=SI). There is no value of I such that 10+I = “2I” [that is, 2*10+I, and 10+I ≠ 20+I].
But, it is possible for (1*10+I) to equal (10+I) --- duh!
So -- S must equal 1.

To maximize SIGN, we choose large values of P and I: P=9 and I=8. This works because a carry of y=1 is possible.

We now have:

   1  1  x 

      9   E  N

  +  8  N  K
= = = = = = = = =

   1  8  G  N

We may still use digits 0, 2, 3, 4, 5, 6, 7.

The maximum yG is 13 (that uses E=6, N=7; make N maximum, not (E+N)=13).

Now, PLZ realize that a value of K=10 is required for (N+K) to equal (1N) {that is, x=1}. So, we conclude that K=0 and x=0.

 

      9  6  7

  +  8  7  0
= = = = = = =

  1  8  3   7

Answer 2

Many answers 

 

Here's one:

  948

  380

--------

1328

 

S has to be 1.

K has to zero

 

E + N must total 12 or more

 

1 + P + I = I + 10

So P is 9