IBM Research | Ponder This | August 2013 solutions

# Ponder This

## August 2013

<<July August September>>

To solve this month's challenge, you can try various types of searches, but a nice and elegant solution is to use the following linear matrix A:

100001101
010001011
001000111
000100001
000010001

and assign each nine-bit vertex v the five-bit value of the result of the matrix multiplication Av.
It is easy to show that the first (v=0) vertex has all 32 types of neighbors. Lets call them v_0, v_1,..., v_31.
From this follows that if we want to go from any vertex w of color c_1 to a neighbor of color c_2, all we need is to take the w xor v_{c_1 xor c_2} vertex.

Don Dodson suggested a nice solution that not only contains the name and date but also the word "IBM" (look down and you'll find it vertically).

PONDERTHISAUG2013BCFJKLXQVWMYZ45
VQMWZY54B3FCKJXLSIUA2G10OPDNREHT
FCB3XLKJMWVQ54ZYDNOPHTREUASI102G
AUIS01G2NDPOTHERWMQV45YZCF3BLXJK
45YZWMQVLXJKCF3B01G2AUISTHERNDPO
HTREDNOP102GUASIXLKJFCB354ZYMWVQ
2G10SIUAREHTOPDNZY54VQMWKJXLB3FC
JKLX3BCFYZ45QVWMERTHPONDG201ISAU
UASI102GDNOPHTREMWVQ54ZYFCB3XLKJ
CF3BLXJKWMQV45YZNDPOTHERAUIS01G2
QVWMYZ453BCFJKLXISAUG201PONDERTH
OPDNREHTSIUA2G10B3FCKJXLVQMWZY54
KJXLB3FCZY54VQMWREHTOPDN2G10SIUA
G201ISAUERTHPONDYZ45QVWMJKLX3BCF
THERNDPO01G2AUISLXJKCF3B45YZWMQV
54ZYMWVQXLKJFCB3102GUASIHTREDNOP

Matej Kollar wrote a nice game that displays the nine-dimens-dimensional cube and helps you color it.
https://github.com/xkollar/PonderGameAugust2013

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com