A Dango Cryptogram+ Clannad Fanfiction?

$\begin{array}{r} D\quad A\quad N\quad G\quad O \\ \times\qquad K \quad Z\quad O\quad K \\ \hline D\quad A\quad I\quad K\quad A\quad Z\quad o\quad K\quad U \end{array}$

From the hint that DANGO has a factor of 5, we can deduce that it ends either in 0 or 5. If it ends in 0, then the U of DAIKAZOKU will be 0 as well, a contradiction. So, O is 5 and U is 0.

After establishing this, since 5 is unavailable anymore, there are only 3 possible single-digit strictly increasing arithmetic progressions, namely (1,2,3,4), (6,7,8,9) and (2,4,6,8).

We can straightaway eliminate the sequence (6,7,8,9). Note that the integer represented by DANGO is a product of 5 and another 4 digit prime number, which is smaller than $9999 \times 5 = 49995$ . There is no way the product can be a 5 digit number starting with 6.

Now, we suppose that D,A,I,K represent 2,4,6,8. We deduce that the first K in KZOK must be 9. If not, we can never get a 9-digit product starting with 24. (Verify this by calculating the range!)

We now know that the Ks represent two pairs of 8 and 9. It is now apparent that the second K of KZOK represents 8 (for us to get 0 in unit digit after multiplication). Therefore, the second K in DAIKAZOKU represents 9. Our partially solved cryptogram now looks like this:

$\begin{array}{r} 2\quad 4\quad N\quad G\quad 5 \\ \times\qquad 9 \quad Z\quad 5\quad 8 \\ \hline 2\quad 4\quad 6\quad 8\quad 4\quad Z\quad o\quad 9\quad 0 \end{array}$

From all this information, we can deduce what G is. Note that $8G+4+25 \equiv 9 \pmod{10}$ . The 4 comes from carry over of $8 \times 5$ while the 25 comes from $5 \times 5$ . This gives us the congruence $8G \equiv 0 \pmod{10}$ . Solving this congruence, we find that G is either 0 or 5, an impossible situation since they are already represented by O and U.

So, our original guess that D,A,I,K represents 2,4,6,8 is false. Now, suppose that D,A,I,K represents 1,2,3,4. As the reasoning before, O and U represents 5 and 0 respectively. The first K of KZOK must represent 9 (check the range yourself!) for the 9-digit product to start with 12. From this, we deduce that the second K in KZOK is 4 while the second K in DAIKAZOKU is 9. Our partially solved cryptogram looks like this:

$\begin{array}{r} 1\quad 2\quad N\quad G\quad 5 \\ \times\qquad 9 \quad Z\quad 5\quad 4 \\ \hline 1\quad 2\quad 3\quad 4\quad 2\quad Z\quad o\quad 9\quad 0 \end{array}$

Same as before, we can now solve for G. Setting up the congruence as before yields $4G+27 \equiv 9 \pmod{10}$ or $4G \equiv 2 \pmod{10}$ . The only possible candidate for G is 8 (out of 6,7,8).

The remaining alphabet N and Z must represent 6 and 7. To decide, we check the prime decomposition of 12785 and 12685. Note that $12785=5 \times 2557$ , 2557 being a prime while $12685=5 \times 59 \times 43$ . Therefore, N represents 7 while Z represents 6.

Finally, a simple computation shows that the wild card little "o" is $\boxed{3}$ . Q.E.D.

I had a lot of fun writing this question. I hope that you enjoy solving it as well. :)