Problems & Puzzles: Puzzles

After knowing the Heinz rectangles it’s easy to imagine the "Heinz pyramids" or the "Heinz stair-cases".

See the following example and the biggest I have found:

Remember: for saving space in this case all primes are of the following form: 12606XXXX

4739 =126064739

4739+4747+4751= 378194237

4739+4747+4751+4781+4801= 630323819

4739+4747+4751+4781+4801+4849+4853= 882453521

4739+4747+4751+4781+4801+4849+4853+4891+4901=1134583313

4739+4747+4751+4781+4801+4849+4853+4891+4901+4903+4907= 1386713123

4739+4747+4751+4781+4801+4849+4853+4891+4901+4903+4907+4937+4969=1638843029

4739+4747+4751+4781+4801+4849+4853+4891+4901+4903+4907+4937+4969+4999+5011=1890973039

4739+4747+4751+4781+4801+4849+4853+4891+4901+4903+4907+4937+4969+4999+5011+5021+5033=

2143103093

Find a bigger one than this:

On December 2005 J. K. Andersen wrote:

The puzzle shows the smallest solution with 9 prime sums.
The smallest with 10 starts with 32485822049.
"32485822" is omitted before all 3-digit numbers here:
049 = 32485822049
049+051+079 = 97457466179
049+051+079+099+129 = 162429110407
049+051+079+099+129+163+183 = 227400754753
049+051+079+099+129+163+183+217+253 = 292372399223
049+051+079+099+129+163+183+217+253+319+321 = 357344043863
049+051+079+099+129+163+183+217+253+319+321+333+351 = 422315688547
049+051+079+099+129+163+183+217+253+319+321+333+351+441+469 = 487287333457
049+051+079+099+129+163+183+217+253+319+321+333+351+441+469+517+553 = 552258978527
049+051+079+099+129+163+183+217+253+319+321+333+351+441+469+517+553+561+583 = 617230623671

The smallest with 11 and 12 sums start with 3490546478609.
"349054647" is omitted before 4-digit numbers:
8609 = 3490546478609
8609+8611+8663 = 10471639435883
8609+8611+8663+8669+8689 = 17452732393241
8609+8611+8663+8669+8689+8723+8743 = 24433825350707
8609+8611+8663+8669+8689+8723+8743+8789+8791 = 31414918308287
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869 = 38396011266013
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971 = 45377104223863
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971+8987+8989 = 52358197181839
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971+8987+8989+9053+9127 = 59339290140019
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971+8987+8989+9053+9127+9193+9209 = 66320383098421
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971+8987+8989+9053+9127+9193+9209+9283+9293 = 73301476056997
8609+8611+8663+8669+8689+8723+8743+8789+8791+8857+8869+8879+8971+8987+8989+9053+9127+9193+9209+9283+9293+9359+9367 = 80282569015723